LIBRARY OF THE UNIVERSITY OF CALIFORNIA. ClMS / A TREATISE ON THE APPLICATION OF GENERALISED COORDINATES TO THE KINETICS OF A MATERIAL SYSTEM BY H. W. WATSON, M.A. POHMKELT FELLOW OP TRINITY COLLEGE, CAMBBIDGE AND S. H. BURBURY, M.A. PORMEELT FELLOW OF ST. JOHN'S COLLEGE, CAMBBIDOB AT THE CLARENDON PRESS 1879 [A II rights reserved ] r"*" ^ " '■<' ^'. } ! PREFACE. The treatment of the kinetics of a material system by the method of generalised coordinates was first introduced by Lagrange, and has since his time been greatly developed by the investigations of different mathematicians. Independently of the highly interesting, although purely abstract science of theoretical dynamics which has resulted from these investigations, they have proved of great and continually increasing value in the application of mechanics to thermal, electrical and chemical theories, and the whole range of molecular physics. The object of the following short treatise is to conduct the student to the most important results hitherto obtained in this subject, by demonstrations free from intricate analysis and based, as far as possible, upon the direct application of mechanical and geometrical considerations. The earlier propositions contain, for the most part, little that is absolutely original so far as results are concerned, but in the concluding portion of the work 111;? 5):? vi PREFACE. the theory of Least Action and Kinetic Foci has been investigated from a somewhat novel point of view, and in a manner which it is hoped may tend to throw some additional light upon this obscure and difficult subject. The language and notation of Quaternions have been employed in two or three instances, but never to such an extent as to break the continuity of the treatise or to prove a hindrance to the student who is unacquainted with that branch of mathematics. TABLE OF CONTENTS. CHAPTER I. PAOK Definitions and General Theorems . . . .1 Arts. 1-5. Definitions of Generalised Coordinates, Generalised Components of Momentum, Generalised Components of Effective Force, Generalised Components of Force, Generalised Components of Impulse. 6. D'Alembert's Principle. 7. Kinetic Energy. 8. Examples of Generalised Components of Momentum. 9,10. Elementary Propositions. 11,12. Lagrange's Equa- tions. 13. Examples illustrative of Lagrange's Equations. 14. Work done by Finite Forces. 15. Work done by Impulsive Forces. CHAPTER II. Maximum and Minimum Kinetic Energy ... 30 Art. 16. Bertrand's Theorem. 17. Gauss's Least Constraint. 18, 19. Exten- sion of Bertrand's Theorem. 20, 21. Thomson's Theorem. 22-25. Ex- amples of Minimum Kinetic Energy. 26, 27. Kinetics of Incompressible Fluid, with Examples. 28-31. Statical Analogy. 32-34. Electro-Statical Analogy. 35. Electrokinetics and Induction Currents. CHAPTER III. Characteristic and Principal Functions . . . 62 Art. 36. Definition and Properties of Characteristic Function. 37, 38. Jacobi's Equations. 39, 40. Definition and Properties of Principal Function. 41, 42. Examples illustrative of Characteristic and Principal Functions. viii TABLE OF CONTENTS. CHAPTER IV. Stationary Action and Least Action . . . . 72 Abts. 43, 44. Stationary Action. 45-47. Examples illustrative of Stationary and Least Action, with definition and properties of Kinetic Foci. 48-59. Ex- tension to the case of any Material System whatever. 60. Illustrative Example. 61. General determination of Kinetic Foci. CHAPTER V. Appendix . 100 Arts. 62, 63. Vector Components of Momentum. 64 -QQ. The Virial. ERRATA. Page 5, line i, replace the sign + by commas, p. 6, 1. 9 from bottom, read H-Ccog. dii dti p. 13, 1. 5 from bottom, for ~ read -—-' do ^01 p. 13, 1. 4 from bottom, for fx read fids. p. 14, 1. 8 from bottom, instead of comma read + p. 16, 1. 9, /or last Q^ead first. p. 22, after last two lines insert where U is the potential energy. p. 23, 1. 2 from bottom, read +Cod^. p. 24, 1. 3 and subsequent equations, read +C-~-' Cut p. 24, 1. II from bottom, for plane of the axes read fixed plane. p. 25, 1. 8, dele 6. p. 25, 1. 10 from bottom, for rcosxirxj/ and rsinyj/-^ read rcoaxj/yjf and r sin \l/\j/. p. 26, 1. 12, read {a + r cos yj/y. p. 26, 1. 15, for Edt. [Turn over. ERRATA. p. 65, 1. II, for < + E read t-\-^, p. 65, 1. 12, for E read e. p. dd, 1. 5, for ^ read ^^. p. 71, two last lines, insert J on left hand. p. 79, 1. 10, /or IX read VIII. p. 80, last line, for system read particle. p. 87, 1. 3 from bottom, read before the kinetic focus or change of type, p. 94, 1. 3, for would read would generally. p. 94, after line 16 insert : — If however MP pass through a configuration i? at which f —f^ the process would not fail, because we could then prove separately that Action {OM-^ME) > Action Oi?, and Action {OE + EP) > Action OP. Now if S be not a kinetic focus the course OSP passes at S through the * locus' f^-=.f^^ and therefore generally every constrained course from M to P infinitely near OSP must pass through that locus, so that at some point E in it f=if^. Unless then aS be a kinetic focus the Action does not generally cease to be minimum at S. p. 94, 1. 7 from bottom, for so long as the Action from retains the same type, it read the Action from to any configuration reached before the first kinetic focus, p. 94, last line, for change of type read first kinetic focus. ( CHAPTER I. "^ot ^'^y DEFINITIONS AND GENEEAL THEOREMS. Generalised Coordinates. Article 1.] When the position of every point of a material system can be determined in terms of any independent vari- ables n in number,, the system is said to possess n degrees of freedom^ and the n independent variables are called the generalised coordinates. The choice of the particular independent variables is perfectly arbitrary, and may be varied indefinitely, but the number of degrees of freedom cannot be either increased or diminished. In a rigid body free to move in any manner there are six degrees of freedom, and the generalised coordinates most fre- quently chosen in this case are the three rectangular coordinates of some point in the body and three angular coordinates deter- mining the orientation of the body about that point, generally the angles Q^ (f), \j/ of ordinary occurrence in rigid dynamical problems. When the body degenerates into a material straight line the number of degrees of freedom is reduced to five ; and when this straight line is constrained to move parallel to some fixed plane the number of degrees of freedom is still further reduced to four. A chain of n links, in which each link is a material straight line, has in the most general case 2^-f-3 degrees of freedom, and if one point in this chain be fixed the number is reduced to 2w, and we might choose for our generalised coordinates the 2n angles which determine the directions of the links. And so on for many other examples. The n coordinates are very generally denoted by ^i, ^2> • -^n- GENERALISED COMrONENTS OF MOMENTUM. [2. Generalised Components of Momentum. 2.] The complete knowledge of the state of any material system embraces not only its configuration but its motion at any instant. Suppose the velocit}^ of each element of the system to be known, and let it be multiplied by the mass of that element so as to obtain the momentum of the element, and let the infinite- < simal variation hq^ be given to any coordinate q^. Then, ii f be the momentum of the element, and if I be the distance of the projection of that element upon some fixed line to which the velocity of the element is instantaneously parallel, measured from some fixed point in that line, it follows from definition that I is some known function of the ^'s, and the virtual moment of the momentum of the element consequent on the variation hq^ is clearly /"— 8^^, and the sum of such virtual moments for the whole system is (2/"-=—) hq^. The coefficient 2 (f-j-) is called tJie generalised component of momentum corresponding to the coordinate q^. In the actual motion each element, as m, is describing a deter- minate curve such that the length s of that curve measured from a fixed point in it is a known function of the ^'s, and the velocity of m is -^ , therefore the generalised component of momentum of the system corresponding to the coordinate q^ . „ ds ds dt dq^ If x^ y, z be the rectangular coordinates of m^ this generalised component of momentum may be written _, .dx dx dy dii dz dz. at dqr dt dq^ dt dq^ In the language of quaternions, p, the vector from the origin to any element of the system of mass m, may be regarded as a 3-] GENERALISED COMPONENTS OF FORCE. 3 function of the n scalar variables qi,...qni and we then have, if the above-mentioned component of momentum be denoted by jo^, A similar definition applies to the generalised component of effective force of the system corresponding to the coordinate q^ , viz. the sum of the virtual moments of the effective forces of all the particles corresponding to the variation hq^ ; and from reason- ing exactly similar to the above it follows that this generalised component may be written _, .d^x dx d^y dy d'^z dz.^ or again using the notation of quaternions, ^dv- dq/ The following notation will be generally employed in dealing with generalised coordinates : — (i) The coordinates will be denoted by ^j, ^2'"3'n^ as above stated. (2) The differential coefficients of these coordinates with regard to the time t, also called the generalised components of velocity^ will be denoted by ^j, q^^^^qn' (3) The generalised components of momentum will be denoted by i?i,i?2" -i^n- Generalised Components of Force. 3.] Let the material system be acted on by any given forces, and let F be one of these forces. Then if I be the distance of the projection of the point of application of F upon some fixed straight line parallel to the direction of jP, measured from some fixed point in that line, it follows from definition that I is some known function of the ^'s, and the virtual moment of F consequent upon any • T,dl infinitely small variation Iq^ in the coordinate q^ \q F j-hq^, B 2 4 GENERALISED COMPONENTS OF IMPULSE. [4. and the sum of such virtual moments for all the forces acting dl dq. on the system is ^(Fj-\bq^ The coefficient of 8^^ or 2 (i^-p^) is called the generalised com- ponent of force correspondmg to the coordinate q^. If the coordinates of the point of application of F referred to any fixed rectangular axes be an, y, and z^ and if the correspond- ing components of F be X, J", and Z^ this generalised component of force becomes ^ d^^^dy_^^a^ dq^. dqr dq^. Generalised Components of Impulse. 4.] When the forces in action are very large^ and the time during which they act is very short, they are called impulses, and are generally measured by the time integrals of the forces. If F be any impulse measured in the manner just described, and if X, J^ Z be its rectangular components, and if /, ^r, y, z have the same meanings as in the last article, then the general- ised component of impulse corresponding to the coordinate q^ will be dl dr du dz ^ F ~ ov ^ (X ^ + Y ^ + Z ^). dq^ dqr dq^ dq^ It must be carefully remembered that the virtual moment of an impulse does not, as in the case of finite forces, represent work done consequent on the variation hq^. When an impulse is spoken of as the time integral of a force, it is only in a particular case that the term is used with strict accuracy, namely when the direction of the infinitely large force which acting for an infinitely short time produces the impulse remains the same during that short time. In such a case, if P be the large force, F the impulse, and r the short time of action, F is accurately equal to / Pdt. But it is quite conceivable that P, the constituent force of the impulse, should vary in direction as well as intensity during the time r. In this case we cannot say that F = j Pdt, but we must say -'0 5-] GENERALISED COMPONENTS OF IMPULSE. 5 that F is the resultant of all the momenta P^dt^^P^cU^^kc, added throughout that interval. In the former case, where the force is fixed in direction during the time r, we obtain the generalised compone'nt of impulse, as we have said above, by writing I Pdt for F in the expression dq In the latter we can only obtain this generalised component by resolving P in fixed directions during the time of its action, and thus we are restricted to the expression Vo dqr Jo dqr Jo dq/ X, F, and Z being the rectangular components of P at any instant. 5.] The terms generalised components of momentum, force, impulse, are very convenient for use, but it is important to remember that they are frequently only names, and do not represent actual forces, impulses, or momenta, still more rarely are they component forces, impulses, or momenta in the ordinary meaning of the term, i.e. such that their simultaneous action or existence is equivalent to the forces, impulses, or momenta acting on, or existing in, the system. For example, let the system of impressed forces be two parallel forces i^and —F at right angles to an axis and at the distances a + b and b from it, and let one of the coordinates be the angle 6 between a plane fixed in the body containing that axis and a similar plane fixed in space. The virtual moments of the forces consequent on the small variation b0 are F(a-\-b)h6 and —Fb,b6, and the generalised component of force corresponding to 6 is Fa, that is to say, it is a couple and not a force. Or again, suppose we are considering the case of a single material particle referred to axes Ox and Oy at the angle a, and acted on by impressed forces parallel to these axes equal to Xand Z respectively* If the coordinate x be varied by bx, the virtual velocities of 6 GENERALISED COMPONENTS. [6. X and Y will he bcc and 6^ cos a respectively, and the sum of the virtual moments of X and Y will be (X-}-Ycosa)bx, so that the generalised component of force corresponding to x is X+ Ycosa, and similarly that corresponding to y is Z+Zcosa ; the generalised components are therefore, in this case, forces but not component forces. Again, suppose that the motion of a particle m is referred to the last-mentioned axes, and that the velocities parallel to a? and y are u and v respectively. Then it will follow by similar reasoning that the generalised components of momentum are m{u-{-vcosa) and m{v-\-ucosa) respectively, i.e. they are not component momenta in the ordinary acceptation of that term. A very interesting example of generalised components of momentum is aflPorded by the case of a rigid body moving about a fixed point and referred to the ordinary angular coordi- nates 6, (^, xjf. Let A, JB, and C be the principal moments of inertia about the fixed point, and w^, o)^, Wg the angular velocities about the principal axes. To find the generalised components of momen- tum corresponding to ^, (/>, yfr, we must resolve the couples of momenta Aca-^, -^^2? ^^z ^^^<^ three pairs, so that one out of each pair has its axis coincident with the axis of 6, (/>, or yjr as the case may be, and the remaining couple of each pair has its axis perpendicular to the axis of dj (f), or xj/. Then neglecting the second couple of each pair we have the required generalised component of momentum. For the coordinate 6 this is found to be by obvious resolution Boj^coscf) + A CO j^ Bin ^. For (^ it is -J- Cco^. For \j/ it is O^cogCOS^ + (Bco^Bmcj) — A(o^cos(f))sm0. 6.] Any material system may be regarded as a collection of discrete particles whose positions are constrained to satisfy certain geometrical conditions by means of internal forces acting amongst themselves. D^Alembert's principle asserts that when its efiective force reversed is applied to each particle, there will be equilibrium among the impressed and reversed effective forces throughout 7-] d'alembert's pkinciple. 7 the system, and that the internal forces above mentioned are in equilibrium among themselves. That is to say, that if the system be slightly displaced, with due regard to the geometrical conditions, the sum of the virtual moments of the internal forces taken throughout the system will be zero, and may be neglected in forming the equation of virtual velocities. If F^ and E^ be the generalised components of impressed and effective force respectively corresponding to the coordinate ^, D'Alembert's principle asserts that the summation being for all the ^'s. , And in like manner, if P^ be the generalised component of impulse acting on a system at rest, corresponding to the co- ordinate q^ and if jp^ be the corresponding component of momentum in the motion caused by the impulse, D'Alembert's principle asserts that 2P ^^ = 2p ^q. Hence, since the 'bqs are independent, we know that the generalised component of effective force corresponding to any coordinate is equal to the corresponding generalised component of the impressed forces acting on the system, and also that the generalised component of momentum corresponding to any coordinate is equal to the corresponding generalised component of the impulses by which the actual motion might be produced in the system previously at rest."^ The Kinetic Energy. 7.] If the mass of each particle of a material system be mul- tiplied by the square of its velocity, one half the sum of the products thus formed, taken for the whole system, is called the kinetic energy of the system, and is generally denoted by the * When the system is subjected to any constraints we may either regard it as a new system altogether, with a fresh set of generalised coordinates fewer in number than before, or we may regard it as being still the same system but acted on by additional constraining forces, such that the sum of the virtual moments of these forces vanishes when the displacements are effected with due regard to the ad- ditional constraints. In this case the equations ^E5q = -SFdq and ISpdq^SPSq will no longer be true for all values of the 5qs but only for such values as are consistent with the constraints so imposed upon the system. 8 KINETIC ENERGY. [7. symbol T. When the generalised coordinates are geometrical magnitudes, lines, angles, and the like, this quantity T may always be express^ed as a homogeneous quadratic function of the component velocities ^1, ^25 ^^-i ^^^ coefficients which are known functions of the ^'s. For each element, as m^ describes a determinate path such that 5, the length of that path from some fixed point in it, is a known function of the ^-'s ; ds n > where the coefficients ^— , -7^, &c. are known functions of the q's and are independent of the ^'s. Hence (-^) , and therefore J Sm^—) or T, must be a quadratic function of the ^'s with coefficients known functions of the ^'s. Again, we have seen that ^ ds ds ^ dt dq,. ds and if the value last given of -j- be substituted in this equation, it follows that p^ is a homogeneous linear function of the ^'s witn coefficients known functions of the ^'s ; and the same being true of each of the p*s, it follows that each of the ^'s is a homogeneous linear function of the ^s with coefficients known functions of the ^'s. Since 7^ is a homogeneous quadratic function of the ^'s, and since each ^ is a homogeneous linear function of each p, it follows that T may be expressed as a homogeneous quadratic function of the p's with coefficients known functions of the ^'s. When T is thus expressed in terms of the jo's it is usually written T^, and when in terms of the ^'s it is written T^l. It may, however, happen that the equations by which the configuration of a system at any instant is determined contain the time explicitly. In such cases the time itself, t, may be .taken as one of the generalised coordinates. 7-] KINETIC ENERGY. 9 Or it may happen that these equations contain not only the magnitudes q^^ q.^^ &c., but also the velocities q^^ q.^, &c. The case in which the time t is one of the generalised coordinates may be illustrated by two particles connected by a rod which expands uniformly, or according to any other known law of time, under the influence of heat. The case in which the ^'s occur as generalised coordinates may present themselves in problems dealing with rough surfaces rolling one upon another, in which the equations expressing the equality of the velocities of the points of contact cannot be readily integrated. In all these cases, as in the simplest case first mentioned, the following equations will remain true : . , ds ds . ds . ds . ds ds (3) p.. = 2m--. But inasmuch as in the case of the time entering as one of the coordinates the corresponding q becomes unity, and in the case of any of the component velocities q^, q.^, &c. so entering ds the coefficients of the type -j- are not all of them independent of the ^'s, it will no longer be true that T may be expressed as a quadratic function either of the ^'s or ^'s. The notations T^ and T^ are sometimes employed in the case where the time enters into the connecting equations ; in these cases they are not quadratic functions as above^ but they indicate the value of the kinetic energy expressed in terms of the co- ordinates and ^ or^? respectively.^ In what follows, w/iere the contrary is not expressly men- * If the time (<) were expressed by any symbol as qt in the connecting equa- tions, and the kinetic energy found on the understanding that qt was to be replaced by unity, then T would, before such evaluation of qi, be a quadratic function of all the component velocities q^, ^2, ■■■ qn ^^^ ?<• The statement in the text refers, of course, to the expression for T in the ordinary form, i. e. after the evaluation of <7t. 10 EXAMPLES OF GENERALISED [8. tioned, it is to be understood that neither the time nor any of the component velocities enter into the geometrical equations of connection of the system. " 8.] In order to obtain actual expressions for the generalised components of momentum in terms of the velocities and co- ordinates in any particular case, it is generally most convenient to employ the following formulae. It is proved above that ^dx dx ^ dy dy dz dz _ ^ \^dx dx dy dy dz dz ] { dt dq^ dt dqr dt dq^ ) Also dx ~dt dx . dx . dx dqn And similar expressions hold for -Sr and -r- dt dt Hence by substitution, „ \ dx dx dy dy dz dz ) ( dq^ dqr dq^ dqr dq^ dq^ ) +(|;)^+01^' ^ c» Example 1. An inextensible string passes over a fixed pulley A. To one end is attached a weight m^^ to ^ the other a moveable pulley C-^. Over the moveable pulley passes another inextensible string having at its ends weights m^ and %. The pulleys and string are supposed to be of inappreciable mass. If the strings hang vertically where not in contact with the pulleys, the system has two degrees of freedom, and we may take for generalised coordinates q^, the length of the first string from the vertex of A to %, q<^ ^^ length of the second string from the vertex of C^ to %. If then a?i, oik-^^ x^ be the heights of %, m^^ and m^ respectively above a fixed plane, we have A 4 6 ^» Fig. I. 8.] COMPONENTS OF MOMENTUM. 11 f ^ = 0, dx^_ dqr ' Therefore dx^_ ^ ^ { ,dx .2 . dx dx . ) = ( Wi + m.-, + mg) q^ + (mg — m^) q^ , and _P2 = (^3 - m^ q^ + (mg + m^) q.^ . If for the weight m^ we substitute another moveable pulley C^ over which passes another string supporting a weight % and a third moveable pulley Cg, and so on till there be A— 1 move- able pulleys, the last supporting two weights m^ and m^^^, we shall clearly obtain the following relations Xr = q^+...-}-qr-i-qr + b for all values of r from 1 to A inclusive, and where b and c are constants. Therefore y^ = 1, — 1, or 0, according as r is greater, equal to, or less than 5, for all values of r from 1 to A inclusive, and —p:^ = 1 for all values of s from 1 to A, and therefore p^ = (2,^+im) ^, + (Sg^+^m-m,) q, + {2,^^'m-m,) q, Escamjple 2. Motion of a chain of A equal uniform links each of length a in one plane, moveable in its own plane, and having one end fixed. This system has A degrees of freedom, and we may take for generalised coordinates 6^, ... 0^, the angles made with the axis of X by the successive links beginning from 0. 12 EXAMPLES OF GENERALISED [8. If T be the distance of an element of the n^^ link from the end of that link adjoining" the next preceding, then for such element, X — 2^"-^ a cos ^ + 7- cos ^^, y — S^""^ a sin ^ + r sin ^^ ; therefore for an element of the first link, dx . ^ dy dx dy For an element of link m^ —— = —a^mOn, or — rsin^,i, or 0, according as m is greater, equal to, or less than n. Similarly, -— — is a cos On, r cos 6.., or : dS^ then if pe^ be the component of momentum corresponding to 0-^^ , we have + --------- + (A-2 + i)a^cos(^,-^2)^2 + - - - - P02 = (^ - 2 + i) «' cos {6^ - e,) 6, + (A- 3 + i) «3 cos'((92-^3) 4 + - - - - and 2>0^ = 3 «^0a- Eoaam^ple 3. The motion of a heavy tube in the form of a plane curve moveable in its own plane, and of a particle of mass m moveable in the tube. This system has four degrees of freedom. Let us take for generalised coordinates, ^15 ^1 ^^® rectangular coordinates of a point A fixed in the tube ; 8.] COMPONENTS OF MOMENTUM. 13 6 the inclination of fclie tangent at A to the axis of a? ; ' (j)^ the angle made by the radius vector from A to the particle with the tangent at A. Let r be the radius vector from A to a point in the tube ; Ti be the radius vector from A to the particle ; (j) the angle made by the radius vector to a point in the tube with the tangent at A ; then r =f{(f) is the polar equation to the curve referred to (Is A as pole, from which r and -j— ^^'^ known in terms of <^. If X, y be the rectangular coordinates of an element of the tube^ cc = 0?! + r cos (^ + (^), y — y^Jf-r sin (^ + ) ; and in like manner for the particle, x = x^-\- 7\ COS {9 + (j)^), 2/ = 2/i + n ^i^ {^ + 4>i) ; then for an element of the tube, ^ = 1 ^ = — = ^^=1- dx^ ' dx^ ' dy^ ' dy-^^ ' — =-rBm(e + 4>), -A^rco^{d + c\>); For the particle, dx • //I , i\ <^.y c^iTj ' dx^ ' c^2/i ' %i ^ = -r, sin(a + (|)i), ^ = ^-i cos {d + (\>,) ; ^=-r,sin(^ + (|>,), ^=ncos(^ + i)j^ — mr^ sin (^ + i)<^i ; 14 ELEMENTARY PROPOSITIONS. [9. + I iJLr COS {6 -\- cf)) — ^ d(f) + mr^ cos(^ + <^i) io P0 = — j fjir sin {d + (f)) — d(f) -{■ mr^ &m {0 -{■ (f)^) x^ + \ //x7'Cos(^ + (/)) — i(^ + mriCOs (d + (/)) 1^1 + ] / /^^^ T^ ^^ + ^^1^ [ ^ + mr/(^^ ; jt;^ ='-mr^ sin (^ + (^i)aci + mri cos (^ + (^i)2/i + wri^^ + wiri^(^i. The integrations are of course from end to end of the tube. 9.] We now proceed to prove certain propositions easily deduced from the foregoing definitions. Proposition I. -—^~p dqr Since 2T=^m(^y = ^mv^; dt T» ^ ds ds . ds . ds ds ds . ds since the coefiicients -r~, &c. are independent of the ^'s and dq^ the ^'s remain constant. If now q^ alone varies, the remaining ^'s being constant, 5 r becomes -^bq^; dq, dT ds dq, ^ dqr or -r-r^ = 2mv— — = p^ by defiixition. dq^ dqr ^ And this proposition is true whether the time enters explicitly into 9.] ELEMENTARY PROPOSITIONS. 15 the connecting equations or not. Remembering that T{^ is in such a ease no longer a homogeneous quadratic function of the ^'s (see Art. 7), but is the expression for the kinetic energy in terms of the ^'s and ^''s. Proposition II. 2 2^ = ^^^^ + ^2^2 + ^c. + i?„^„. Since I'^ is a homogeneous quadratic function of the ^'s, it follows that But -^~V\' -r^ = v%' &C-; .-. 2T = p^q^ +PA2 + &c. + _p„9„. When T' is expressed in terms of the p's and ^'s, as in this proposition, it is written T^^. Proposition III. -~ = Qr, and -^ + — — ^ = 0.* dpr dqr ctqr Since T^^^ T^, and 7^ are three different expressions for the same magnitude, and Tp^ = ^I^pq, .-. T, + T^ = ^:pq. Let all the variables j9, q, and q be varied, then But by Proposition I -j-r- = A 5 Now the 2n magnitudes /'i-.-j^n and ^i...^„ are independent, and therefore their variations 8i?j...8i?„ and 8^i...d^„ are in- dependent ; ^ dTp_ , ^^^ ^ ^ - c?^,. c^r/,. dqr * This demonstration is taken firom Maxwell's ' Electricity.' 16 ELEMENTAllY PROPOSITIONS. [9. In the foregoing demonstration it is assumed that the time does not enter explicitly into the connecting equations, and therefore that T^., and T^ are homogeneous quadratic functions of the ^'s and ^^s respectively; in which case, as has just now clT been proved, the equation — ^ = ^^ follows as an analytical con- sequence from the proposition —~ = p^. It may however be demonstrated independently, and tohether the time enters into the dT connecting equations or not, that —^ = ^,., it being bprne in mind^ as in the last Proposition, that T^ ceases to be a homogeneous quadratic function of the j!?'s in such a case. ^ ^ ,dx dx dy dy dz dz. For -p^ = 2^( _l_._^^4. ). dq^ dt dq^ dt dq^dt Let a small impulse act on the system, whereby the velocity of each element is varied without change of position. „,, ,, .... dx dy dz .„ - ... dx dy dz Then the quantities -77' -77' -77 will vary, while -?-> -i^> -r- ^ dt dt dt -" dq,. dq^ dq,. remain constant ; . ^ V ,dx dx dy dy dz dz. dq^ dt dqr dt dq^ at' . . 5> V /• dx dx . dy dy . dz ^dz. ^"^ ^' <^^r dt ^"^dqr dt ^"^dq,. dt' ' 'SiqOjp „ (, . dx . dx „ .^dx , . dy . dy , ^dy . dz ^ , dz „ . ^dz , /. az dz ^ ^ .^dz) dq^ ^^dq^ ' dt) ^dt dt dt dt dt dt' d'p And the 8jo's being independent^ . ^_ • 9-] ELEMENTARY PROPOSITIONS. 17 Proposition IV. -yr = , r- Since T^ is a homog'eneous quadratic function of the ^'s it must contain a term of the form Cq^q^, where C is independent of the ^'s. dT- .'. pr = -1-r- = Cq^ + terms independent of q^ . dT- Similarly p^ = -—■ = Cqr + terms independent of q^ ; dq, dqr Proposition V. If pq and p'q' represent two different states of motion of the syste^n in the same configuration^ ^pq = Sjp'^. If variations bq-^, hq^^ &c. of the coordinates give rise to the displacement 8r in any particle m whose velocity is v^ we know from definition that SpSg = ^mvhr cos a, where a is the angle between v and hr. When the system is in the p, q state, let the variations 8^1, hq^i &c. be given to the coordinates such that Hi = ii ^* > Hi = ^2 ^^^ ^^• Then it follows that br = v'bt, .'. 2j)^8i = Swivv'cosaS^. Similarly ^p^qbt = Smv'v cos a8< ; .-. ^pq' = :2p'q. This proposition may also be deduced from the linear equations connecting the jo's and ^'s. Proposition VI. Ifpq,p'f represent momenta and velocities i?i two distinct motions of a system in the same configuration^ then if q + q' represent velocities in a third state xf motion with the same configuration^ p-\-p' ivill represent the momenta in this third state. This follows from the fact that the j^'s are homogeneous linear functions of the ^'s with coefficients known functions of the ^''s. 18 ELEMENTAKY PROPOSITIONS. [9. PROPOSITION VII. If ^5 / and q -\- q represent coi velocities of any system in the same configuration hut different states of motion^ and if the notation T^ represent kinetic energy corresponding to the state q, Since T^^ is a homogeneous quadratic function of the n varia- bles ^1, ^2 &c., it follows from Taylor's theorem applied to any number of variables that where R is independent of the ^'s and therefore by symmetry must be equal to T^. Also aJ-q aq=^'^ .'. Ti^i.= Ti + T^y-\-^pq\ = T^+T^ + :2/q. Similarly T^_^=T^ + Zr-:^pq', = T, + T,-^/q; with similar propositions concerning T^^^> and T^_^.. Proposition VIII. If ^pq = 2//, then either is greater than ^p'q or 2jo^'. For if ^ — q represent the velocities in a third state of motion, p — p' represent the momenta in that state by Prop. VI. Therefore i2(jt? —p')[q -q^) represents the kinetic energy in that state and is therefore positive ; that is 2pq + ^p^q — ^p^q — ^pq is positive ; that is 2 ^pq — 2^p^q is positive ; and therefore ^pq or ^p^q is greater than ^p^q or ^pq. It appears from this that ^pq—^pq^ can never be zero unless q = q' for each coordinate. lO.] ELEMENTARY PROPOSITIONS. 19 10.] If /'($'i ••• $'«), or shortly/, be any function of the co- 7 -fi ordinates, and — or /" its rate of increase per unit of time, then, if f be given, T is the least possible, and if T be given, f^ is the greatest possible, when for each coordinate p is proportional to -j- • For let q, p he a. set of velocities and momenta in a motion such that ^q-4-y or -77, has the ffiven value, and that p=:\-^, ^ dq dt ° ^ dq where A. is some constant. Let ^ + /, JO +y be the velocities and momenta in another 7 J} motion which gives the same value for ~- > and therefore such and therefore ^q^ -7- = 0. ^ dq Then for the doubled kinetic energy of this second motion we have 2T=:2(p+p^(q + q'), = 2^^ + 2/^ + 22^9^', = 2j02 + 2jpY+2A2^'^, = 2p^ + 2/2'. which exceeds '2pq, the doubled kinetic energy of the j», q motion, by 2jf?Y, an essentially positive quantity. In order to find the actual value of this least kinetic energy, we must express every q in terms of the jo's in the equation f-^^Tq' /J -f and then substitute A. -7- for p. dq ^ If the linear equations expressing q in terms of the ^'s be of the form c 2, 20 ELEMENTARY PEOPOSITli the result is /= = x|^„(gf-.: 2« '^^'^f = = \F, suppose ; whence k: = ^ and 2^=5 [lO. 7 -/* i'' is the expression for 2 7^^ with -j- written for p. Secondly let T be given, and let q, p be a set of velocities and momenta in a motion such that 2^ has the given value, df and that p = X-f-t and let q + q\ p +y be those in any other motion having the same kinetic energy. Then 2^+/ ^ + ^' — 2jp^ = 0, or 2jpY + 22p2' = (1) Therefore Now let q' denote a set of velocities proportional to the q set so that q •=. rq\ where r is some numerical quantity, and con- sequently y = rp^\ Let further r be so chosen that 2/'^" = Ispq. Then from (1) 2 2;;^" r = = — —' And (2) becomes II.] Lagrange's EQUATIONS. 21 which is necessarily positive by Proposition VIII. Therefore (2 ^if or f is greater than (S ^(j + j'))^. 11.] If p^ and F^ be generalised components of momentum and force corresponding to any coordinate q^, then at dqr We have seen in Art. 2 that .div dec dy dy dz dz , > ^' = ^"^^W^r + ^"^ + dt'd^)' ^*' „ ^ ,d^x dx d^y dy d^z dz. ,_. and that ^r = 2m(-^.-^ + -z|.-r^ + -r^.-^)- (2) ^ ^dt^ dqr dr dqr dr dq/ Therefore by differentiation of (l), remembering (2), dPr „ V, id^ d dx dy d dy dz d dz. dt ^dt dt dq^ dt dt dq^ dt dt dqj Now d dx . d^x . . d^x , ^ d , . dx ^ . dx . d dx ~ dqr dt , ,. , ^ d dy d dz andsim-larlyfor ^^ and ^^^y, provided that in differentiating with regard to qr the ^'s remain constant ; ,'. —J- = I^r + -J » dt dqr dpr dT,i __ ^ ^'* ~dl~~d^r~ '' dT- Since, by Proposition I, -^ = Pr> ^^is result may be written in the form ±.l^\^^ = Fr. dt ^ dqr dqr 22 EXAMPLES OF LAGRANGE's EQUATIONS. [l2. And since the demonstrations of this proposition as well as that of Prop. I hold when the time enters explicitly into the con- necting equations, we have hi all cases dt dqr dt'^dqr dq *■' remembering that T^ ceases to be a homogeneous quadratic function of the ^'s when the time thus enters. By Proposition III it follows that when the time does not enter explicitly the result may be written dt dq,. dt dq^ *"' 13.] These are called Lagrange's Equations of Motion. They are applicable to systems moving under the influence of finite forces only. The corresponding forms for impulsive forces are easily deduced from the foregoing propositions. For if P be the generalised component of impulses acting on the system cor- responding to the coordinate ^, we have seen that if the motion be from rest P = jo. But the velocities created in the system by any impulses are irrespective of the state of the system as regards rest or motion at the time when the impulses act. Therefore if /?o denote the momenta before, and ^ after, the impulses, q^ and q the cor- responding velocity components, we must have P = p-Po, dT dT „ dq dq^ 13.] The following are examples of the use of Lagrange's Equations. Eocamjple 1. The system of two pulleys in Example 1, Art. 8 moving from rest under the action of gravity; it is required to determine the motion. Evidently in this case dT ^ dT therefore dq. = 0, ^^2 = 0; dp^ dU _ dt ^^ dq. = K- -m^-m,)g, ^Ih dU / \ dt ^ """ di=-(^-l)5'«> (A-2)^, + A^2 + (A-3)^3+-+^A-i=;^2=-(A-2)^<. &c. = &c. From which any q can be expressed in the form of a determinant. Example 2. The following is taken from Routh's Rigid Dynamics. To deduce Euler's equations for a rigid body from Lagrange's equations of motion. "We have shewn above that if _p«, Jp^, p./r denote the generalised components of momentum corresponding to the coordinates 6, <^, y^ respectively, 2)Q = 5(02 ^^^ (/) + -4 coj sin (^, p^— -Ccog, 2)^ = Ccog COS 6-{-{B(A^ b\ji<\)—A «i cos ) sin 0. 24 EXAMPLES OF LAGRANGE's EQUATIONS. [13. But by Lagrange's equations ~dt ^ "~ d^' that is —C —7^ + (-^^2 sin (/)— ^ coj cos (/>) 6, — {Bco^coscf) -{■ Acd^ sin(f)) sin^\fr =__— . But 6 = ot)^sm(f) + 0)2 cos cf), sin ^\^ = — coj cos (^ + CO2 sin^. Hence we obtain by substitution, — — - = —G ~j^ + {B 0)^ sincjy — A (Oj^ cos 0) (wj sin (f) + (jo,, cos <^) — (5 CO2 cos (/) + ^ fOj sin (|)) ( — cOj cos (/> + cOg sin (j!)) and the two other equations of Euler's system are deducible from this by symmetry. Example 3. A rigid body is supported on a fixed axis and another rigid body is supported on the first by another axis. Case (a). If the second axis be parallel to the first. (Thomson and Tait's Natural Philosophy, § 330, p. 257.) Here there are two degrees of freedom, and the coordinates may be conveniently taken to be (i) )] — m'a6 sin (\/^ — <^) \^ = , or (m j^ + m^a^) (f + m' ah "^ cos \j/— (f)—m ah yj/^ sin -^—(f) = 4>, m'a6 — [(^ cos (\/^ -(/))] + m (62 + ^2^ x/; + m' c*& sin (\/a - <^) (^ \p' = ^ ; where and ^ are the generalised force components corresponding to + r^yir''}. The kinetic energy of rotation of the second body is where A, B, C are its principal moments of inertia about the C. G., and fo^, 0)2, CO3 the rotations about the principal axes at that point. ^'q' 26 EXAMPLES OF LAGRANGE's EQUATIONS. [13. The quantities a)j, oi,^, Wg will be linear functions of <^ and \fr with coefficients functions of y\f, which can only be expressed when the circumstances of each particular case are known. The kinetic energy of the first body is \ /^^, where I is the moment of inertia of that body about the first axis. As a particular case, suppose the C. G. of the second body to be situated in the plane containing the first axis and the shortest distance, i.e. suppose that 6 = 0. Suppose also that the second axis is parallel to a principal axis through the C. G. of the second body. Then the kinetic energy of translation of the second body becomes — {(a + r cos\//)<^^ + r^\/r2}. Li And the kinetic energy of rotation of that body becomes i I {^ cos^ (^/a + a) + ^ sin2 (^ + a)} (^2 _j. ^^2 1 . that of the first body being as before \ I^^. Therefore twice the kinetic energy of the whole system assumes the form (P+^cosx/A + i?cos2|/, + >S'cos2^/H^)<^2_,_^^2^ where P, Q^ R, S, and U are known functions of the given constants. And Lagrange's equations become -^.({P + ^cos\// + i?cos2x/A + /Scos2(i// + a)}<^) = 4>, 4> and ^ being generalised components of force corresponding to and \|/ respectively. If the first axis be vertical, and if gravity be the only impressed ^^^^®' 4> = and * = —m'gr cos \/r, and therefore the equations become {P + QcoByjf + Rcos^\lf + Scos^{yl/ + a)}ip = E (const.), E^ ^ {P+Q cosyjf + E cos^ ylf-\-S cos^yl/ + a)) "^ " (P + Qcos^ + RcoB'if + Scos'(-^-^a)f " -"^^roosylr; :,, 2E2 ^^ ^ "^ {P + Q cosy)/ -{-R cos^ x/a + >S cos2 (^ ^ ^)) 2 m^gr (sin fi — sin \j/) = — u ' 1 4-] WORK DONE BY FINITE FORCES. 27 giving "^ ^"""^ =/W; whence t may be found as a function of -^ by mere integration, and therefore conversely \|^ may be found as a function of t. And then by substitution <^, and therefore (^, is found from the equation . _ d^_ E ^ ~ dt " P-^QcoB\l/-{.Rcos^yj/ + Scos^{\j/Ta)' If the circumstances of the motion make a = 0, the expressions are slightly simplified by the two constants 7? and S blending into one. 14.] If F^ denote the generalised component of impressed force corresponding to the coordinate q^., the work done per unit of time by the impressed forces on the system moving with the velocities qi, ...q^i that is the increase per unit of time of the kinetic energy, is ^Fq ; or by Lagrange^s equations, .dp dT^ . Now p=.2q±fr + 2^q; dt ^dt dq dq^' + - - - - and 22— ^ = i2,22-^^ + 4j,22-^+.... Therefore 2fj = 2j(|-^^) If the velocities be indefinitely small the last term may be neglected in comparison with the others, because it involves only higher powers of the ^'s. In that case, but not otherwise, we 28 WORK DONE BY IMPULSES. [15. may equate the coefficients of each q, and obtain for a system at rest, g.) If the velocities ^j , ... q^. only be reduced to zero, the work done per unit of time on the system moving with the remaining velocities q^^^ . . . ^„ becomes by arrangement of the terms r+i^^-?r+i2, -^q + q,^,\ -^ or ^^'-^^^ dq Now let ^1 . . . ^^ define the position of a moving space, S'r+i ••• $'n that of a system moving relatively to the space. In that case the second member of the last equation expresses the increase per unit of time of the kinetic energy of the relative motion. And the equation shews that this is obtained by sub- tracting from any component of force — e. g. F^^^ — the quantity S\ "jt^ q, which is what the generalised component of effective force corresponding to q^^-^ would be if the space were at rest and the system fixed to it, ^V+i ••• $'*n being therefore zero. This expresses Coriolis' theorem. 15.] To find the work done by any impulse acting on a ma- terial system in any given state of motion. Let q, p represent any components of velocity and momen- tum before the impulse acts, and let q + q and p-\-p^ be the corresponding components after the impulse. Let jP be any component of the impulse cori'esponding to the above-mentioned components of velocity and momentum. 15-] WORK DONE BY IMPULSES. 29 Then employing the notation of the preceding articles we know that the work done by the impulse must be equal to or 5^^ + ^^/ + 2/^-Tp, or ^Sp'^' + 2/^. Let the new velocity q^-g' be denoted by ^, then this ex- pression for the work becomes Also by D'Alembert's principle X = -^'• Therefore the work done by the impulse whose generalised components are P^, Pg? ^^-j is If the impulse whose rectangular components are X, Z, Z act at a point of the system whose coordinates are a?, y, z and com- ponent velocities u, v, w, and if ?7, F, W be the values of w, v, w after the impulse, then the work done will be found by substi- tuting X, J, Z for Pj, P2, Pg in the above expression and making each of the remaining components, P4, P5, &c.j zero ; so that it becomes ^.u-^U ^v+F ^w+W CHAPTEE IT. MAXIMUM AND MINIMUM KINETIC ENERGY. ^ Article 16.] If any system at rest in any configuration he acted on by any given impulses, the kinetic energy imparted will be greater the greater the number of degrees of freedom of the system. And for every additional constraint introduced there will be a loss of kinetic energy equal to that of the motion which, compounded with the unconstrained, would produce the constrained motion. (Bertrand's Theorem.) For let P^, Pg' ^c- ^^ ^^® generalised components of impulse acting on the system. Let q^, q^^, &c. be the resulting com- ponents of velocity, />i,/>2 5 ^^- ^^® corresponding momenta, and T the kinetic energy. Then by what has been already proved, we know that Pi = Ih^ Pi = Piy &c., and T=\ ^pq. Let any constraint, which we may denote by C, be introduced into the system, such that when the same impressed impulses act upon it as before, the velocities and momenta in the constrained motion shall be q-^, q^, &c. and p{, p^, Sec, and the kinetic energy T' = ^ Sy/. In the constrained system the possible displacements 'ciq■^, 'bq^, &c. are no longer independent, but it is still true by DAlem- bert's principle, Art. 6, note, that if ^^i, ^^2^ ^c- represent any possible values of these displacements in the constrained system, 2 (P-;/) 2)^ = 0, although we cannot, as in the former case when the Tiqs, were independent, equate the coefficient of each 'bqto zero, and deduce the equations Py=p\, ^2=^25 ^^- ^t is clear that if we gauss' least constraint. 31 take c)^i, c)^25 ^^' proportional to q{, q^', &c., such values will be consistent values of the ^^'s, and therefore or 2^/^'= "EiPqziz ^pq = 2j^/^. Therefore ^' = i 2 j/q = i 2 ;; ^'. And T-T'= 4 {2;;^-2;;f } since ^p'ii-q) = 2/ ^-2/ 3' = 0. That is, T-T'=T^_^. The motion which has to be combined with the free in order to produce the constrained motion, that is the motion q—q, may be called t/ie constraining motion, 17.] It follows as a corollary that the kinetic energy of the constraining motion q —q is less than that of any other motion which, compounded with the free motion, would cause the system to obey the constraint C : in other words, T^_^^, the kinetic energy lost by the introduction of the constraint (7, is the least possible. This is Gauss' principle of Least Con- straint. Let q denote the velocities which the system when subjected to the constraint C, and to no other constraint, actually takes under the given impulses. Let ^" denote the velocities in any motion whatever which the system can have consistently with the constraint C. Then as we have seen, by D'Alembert's principle and therefore also 2(i-^0/' = O.j Then cf -\-cf^ — q represents the velocities in any motion, different from q' — q^ which, when compounded with the free motion, satisfies the constraint C. And we have TM-i- J'^-i = J 2 (p'+ p"- p) (f + j"- j)-i 2 ip'-p) d'- q) = i2(/-p) r+ iS(j'- 2)/' + i2/Y' = i2/Y'by (1); and this is necessarily positive, therefore 32 gauss' least constraint. [17. This proposition sometimes admits of practical application if it be required to find the constrained motion when the free motion is known. If, for instance, only one degree of freedom be removed by the constraint, then the constraint may be ex- pressed by making some one function of the coordinates constant in the constrained, which is not constant in the free, motion. If/" be that function, 2y-^, the rate of increase off per unit of time in the free motion, is known. In the constrained motion f is to be constant, that is Now the kinetic energy of the q'—q motion is, as we have just seen, less than that of any other motion which, combined with the free motion, satisfies the constraint ; that is, in which the rate of increase oif per unit of time is Therefore by Art. 1 0, from which jo'— ^ may be determined as in Art. 10. For example, two free particles of masses, ^j, %, move from rest under given impulses with velocities ^i, yi, ij, ^25 ^2^ •^2' It is required to determine the velocities with which they will move off under the same impulses if constrained to remain at a constant distance, r, apart by being connected by a string or rod without mass. If ^/, &c. be the new velocities, we have And determining A. as in Art. 1 0, we find dr . ^ l8.] EXTENSION OF BERTRAND's THEOREM. 33 where -j- is the rate of increase of r with the time in the free motion ; that is, A since Avr— ^'+^^''^^- fa -^2)' , (2/1-2/2)' ■ k-^2)' = 1, and similarly (^r + (|;r-H(|^r = 1. Hence . , . m, cc, — fl?„ (Zr jp ' j^ ^ f i £ m^-\-m^ r dt &c. = &c. The general problem of determining the constrained motion, when the free motion and the nature of the constraint are known, is more conveniently treated under the principle of least kinetic energy hereafter discussed. For every constraint must act at some definite point or points of the system, and may be conceived to consist in giving to these points certain new velocities in addition to the velocities which they take in the free motion. The kinetic energy of the constraining motion is then, as will be proved presently, the least which the system can have consistently with those points having the required new velocities. And this property, as will appear, suffices to determine the whole motion. 18.] The proposition proved in Art. 16 has been put into a somewhat more general form by Lord E-ayleigh in the PAil. Mag.^ vol. xlix. § 4, which, expressed in the language of generalised coordinates, is as follows. Let Pi, P2, &c. be any generalised components of impulse acting on any material system. Let Qi, §25 ^^' ^® ^^y possible quantities whatever, and let T(i be the value of the kinetic energy of the system, when with the given configuration the velocity components are Q^, C2. &c. 34 EXTENSION OF BERTRAND*S THEOREM. [19. Let the expression ^FQ—T(^ be denoted by the symbol % and let \//- be the "value of 4^ when for Q^^ Q^, &c. have been substituted the values ^j, q^, &c. of the component velocities actually assumed by the system at rest in the given configura- tion when acted on by the given impulses. Then -^ is the greatest possible value of ^. For if Px^ Pi^ ^^' be the momenta actually assumed,, we know that = T^ + Tq-^pQ = T^-Q by Proposition VII, and is therefore essentially positive. The result of Art. 1 6 is a particular case of this proposition. For if ^1, ^2 5 &c. be the velocities assumed by the system when subjected to any constraint and acted on by the same impulses, ^ is the kinetic energy assumed by the system, that is Tq. and the result just obtained assumes the form the same as that of Art. 16. 19.] By the aid of the foregoing we may prove that when the masses of any part or parts of a material system are diminished, the connections and configuration being unaltered, the resulting kinetic energy under given impressed impulses from rest must be increased. Substitute for the ^'s in forming the function ^ for the new system the values ^j, q^, &c. of the velocities assumed under the given impulses in the old system, and let 4'' be the value of ^ thus found in the new system. Also let T'^ denote the kinetic energy of the new system corresponding to the velocities ^1 . . . ^„ . Then A+2-"A ^^^ separately zero, then the resulting motion of the system will he such as to render the kinetic energy the least possible consistent with the given velocities q^, ^2---^r* (Thomson^s Theorem.) For the conditions that the kinetic energy should be a maxi- mum or minimum consistent with the r velocities ^i, ^2-"^r ^^® dT^ ^ i^ ^ ^ = • or Pr+i=Pr+2'-- —Pn— 0- Let ^^^p ^,.+2---^n ^6 the values of the n — r unknown velo- cities determined from these equations, and let T^ be the value of the kinetic energy with these velocities, and let 2\ be its value when any other values as ^r+i + ^'r+i---^n + ^'» 3,re substituted for these n—r velocities, the first r velocities remaining the same as before, then T^ = T^^-T.'+'Lpq, by Prop. VII. D 2 36 THOMSOI^'S THEOREM. [2 1. where T^^ is the value of T with the first r velocities separately- zero and the last n — r velocities fr-^i-'-i^i respectively. Also in ^pq the first r velocities are separately zero and the last n — r momenta are also zero ; .-. 2pq=0. or T,= T,-T,;; that is to say, T^, the kinetic energy determined by the condition that the last n — r momenta are separately zero, is less than the kinetic energy with momenta different from these, and the first r velocities the same as before, by the kinetic energy of the system in which each velocity is equal to the difference of the corresponding velocities in the original and altered system. It follows from this Proposition that whenever a material system in any given configuration is set in motion by impulses entirely of given types in such a way that the velocities of the corresponding types have certain given values, then the motion of the system may be entirely determined by the condition that the kinetic energy assumed is the least possible with the given configuration and given velocities, the number of given equations among the velocities together with the equations of the form p =z being equal to the number of independent variables. 21.] Hence we may deduce the following theorem : — If a material system at rest he set in motion hy any impulses^ the kinetic energy with which it moves off is the least which it can have consistently with the velocities assumed hy the points at which the impulses are applied. For suppose that the connections of the system are such that r of the generalised coordinates are known functions of the 3 m coordinates x^ y, z, &c. of the points of application of the im- pulses and of these variables only. Let the points of application of the impulses be m in number, viz. Oj, Og... 0^^; then the ^m coordinates of 0^ ... 0^ are each of them determinate functions of the r coordinates $'1 , ^2^ • • • ^r • Let X, Y, Z be rectangular components of any of the impulses > ~- i -7— will be separately zero if s > r, because the 2 2.] EXAMPLES OP MINIMUM KINETIC ENERGY. 37 acting- at any one of these m points, then by definition any generalised component of impulse, as ^, will be dq, dq, dqj Then ij will be always zero unless s lie between 1 and r inclusive, for if x, y, z refer to any point of the system other than O^y O.^...O^j the values of X, J, Z are separately zero; and if a?, y, z refer to one of these points, the values of dx dy dz d^J ^ positions of these points are functions of the first r ^'s and of these only. In this case therefore the components of impressed impulses ir+i...i?j are separately zero, and therefore the generalised components of momentum Pr+i-'Pn ^^e separately zero; and therefore if the velocities of the m points, and consequently the values of ^j...^^, are given, the proposition of Art. 20 shews that the kinetic energy must be a minimum with these given velocities. It may also happen that some of the momenta corresponding to ^^ . . . q^ — e. g. p^ and j?^_i — are zero. In that case the kinetic energy is not only the least possible consistently with q^... q^, but also the least possible consistently with q^ ... q^_2 . 22.] The following are examples of the use of these theorems : Example 1. The system of pulleys described in Art. 8 being at rest, let any velocity q^ in a vertical direction be given to the weight m^ by an impulse applied at ?7^/. it is required to determine the initial motion of the system. If there be only one moveable pulley, we have only to make p^ = 0, that is (mg - m^) q^ + (m^ + m^) q^ = 0, m„ — m which determines the motion. In like manner if there be A moveable pulleys, the expressions for p^, p^, &c. given in Art. 8, equated to zero, give as many linear equations as are necessary for determining B8 EXAMPLES OF [23 Example 2. In the chain of X links discussed in Art. 8 let any velocities in the plane of the chain be given impulsively to P, the extremity of the r^^ link from by impulses applied at P. Here, unless r be unity, the system would lose generally two degrees of freedom if P were fixed, and therefore the rectangular coordinates of P might be expressed as functions of two generalised coordinates. In the system of coordinates employed in Art. 8 they are not in fact so expressed unless r — 2. Generally, oc = 2^j a cos d, 2/ = 2*^1 ^ sin 6. In order to determine the initial motion when r >2, we must either first transform the coordinates, or seek by the general method of the calculus of variations to make T a minimum consistently with the given velocities of P, that is with 2''j a sin 6 $ and ^''j a cos 6 0. If for instance the velocity of P in one direction only be given, and be produced by impulses acting in that direction only, we may take the given direction for axis of x, and then we have from 2\ to p,. inclusive 7; oc sin 6, and 2^r+i = . . . ^;^ = 0, from which the ^'s may be determined as in Art. 10. If the velocity of P be given in both directions, or if more than one point be struck, the expressions would assume complicated forms. 23.] Certain very interesting examples of the use of the propositions of Arts. 20 and 21 are given in Thomson and Tait's Natural Philosophy. These will repay fuller discussion. For instance, a rigid body is set in motion by a blow applied at a certain point in such a way that the velocity of that point has a certain determinate value in magnitude and direction. It is clear^ from what we have just now proved, that all we have to do is to express the kinetic energy of the body in terms of the three component velocities of the point struck and three other variables, and to make this kinetic energy a minimum. Let the body be referred to the principal axes through 0, the point struck. Let u^ v, w be the given velocities of 0; coa;, (o^,, w^ the angular velocities round the coordinate axes ; A, B, C the moments of inertia round the axes ; x, y, z the coordinates of the centre of gravity; ilf the mass of the body. Then we have, 24-] MINIMUM KINETIC ENERGY. 39 since the component velocities of any element m of the mass of the body situated at ^, ^, ^ are u + ZiOy — y(Oz, v + xoy^—zdijc, and w + yto^— ajw^, + 2M \cc{voi^—woi)y] +y {'Wbiy.~'^^z} + ^{tta)^ — vcoj.}? ; whence we obtain by the ordinary method, making T a minimum, A(jdy.-{-M [wy—vz] =0, B(i)y + 3I {uz—wx} = 0, Ca)g + M {vx — uy} = 0, which determine ca^, (jdy, co^. We might in this case obtain the same result from the assumption that the moment of momentum round each axis through is zero. 24.] Again, an inextensible string is set in motion by im- pulses applied at its ends in such a way that the velocities assumed by the ends have certain given values. We have to express the kinetic energy of the whole string, paying regard to the equation of continuity which expresses the inextensibility of the string, and remembering to take account of the given velocities of the points pulled. This example is fully worked out by Thomson and Tait, pp. 226-229. We here vary the geometric treatment by intro- ducing the notation of quaternions. It is obvious that the terminal impulses are necessarily tan- gential, since any impulses applied at right angles to the tangent would generate in the extremity of the string an infinite velocity, without instantaneously affecting any other portions of the string. Let IX els be the mass of an element ds of the string at P. Let pp be the vector from the origin to P, pp the vector velo- city of P. Also let pq, pq be the corresponding vectors for a neighbouring 40 EXAMPLES OF [24. point Q of the string. Then, by the condition of inextensibility, T{pp-^Pq) is constant; that is, or S.(pp-pii) (pp - pg) = 0. Thatis, 5.1p^=0; as as or, writing as usual / for j- > ^"1'=" v be an element of the normal. Let K be the density of the fluid at any point. K is therefore an essentially positive quantity. Let os, y^ z be the coordinates of any point referred to rectangular axes. Let F" be a function of x, y^ z satisfying the following conditions, viz. ?=^ dv at every point on the surface, and dx dx' dy^ dy' dz^ dz' ~ at every point within the vessel ^. Let u, V, w be the initial velocities taken by a particle of the fluid. Then a motion in which dV dV , dV * u = -=- , V = -7— , and w = -r- » dx dy dz satisfies the surface condition • %" ■« at every point on the surface, and also satisfies the equation of continuity, viz. d ,^^dV. d ,^^dV. d ,^^dV. ^ ,. "■• ^(^^) + ^(^^) + ;i^(^:sr) = « (2) at every point within the vessel, and is therefore a possible motion of the liquid subject to the given surface conditions. If we can show that it has less kinetic energy than any other motion satisfying the same conditions, it must by our principal proposition be the motion actually assumed by the liquid. * See Maxwell's Electricity, vol. i. p. 104. 2 5-] MINIMUM KINETIC ENERGY. 43 If the actual velocities be not dV dV , dV -ij— } -^— ) and -rr- dx dy dz at every point, let tliem be _+a, — +/3, and— +y. Then in order that the surface conditions and the equation of continuity may be satisfied, we must have a = 0, /3 = 0, y = at every point on the surface, and at every point within the vessel. dV Then the kinetic energy of the motion -j- +a, &c. is + lfJfj^{a' + ^' + y'}dxdydz . fffzr \ dV dV dV) , , , By Green's theorem the third line is equal to n K Va dy dz + [j KV^ dx dz + [(k Vy dx dy and is therefore zero; since a, /3, y are zero on the surface, and the quantity under the triple integral is zero within the vessel. Hence is less than and therefore the kinetic energy of the motion -j- j &c. is less 44 KINETICS OF [26. than that of any other motion satisfying the given surface con- ditions and the equation of continuity. This motion is therefore the actual motion. The process itself shews that there can be only one function, Vj o^ x^ y, z satisfying conditions (1) and (2), except as such function may be varied by the addition of a con- dV stant. Therefore -7- , &c., or %^ v, and w, have single values at every point in the fluid. In other words, for any given initial motion of the containing vessel there is a single determinate motion of the fluid. Evidently — - = — / &c., and the motion is of the kii\d called non-rotaiional. Fis called the velocity potential. The above investigation would evidently apply if, instead of a single vessel enclosing the liquid, there were several vessels, and if the liquid had immersed in it any rigid or flexible bodies bounded by closed surfaces. 26.] In Thomson and Tait's Natural Philosophy the use of generalised coordinates is illustrated in a very interesting man- ner by their application to certain cases of fluid motion. Given an incompressible homogeneous fluid, either infinite in extent or bounded by any finite closed surfaces of any form, and with any rigid or flexible bodies moving through it, it may be proved that the kinetic energy of the whole fluid is known at any instant if the velocities of the containing surfaces and those of the moving bodies are known. This truth can be established by some such reasoning as follows. It is true that although the positions of the containing surfaces and immersed bodies be known, the system has in re- spect of the relative motions of the particles of the fluid a prac- tically infinite number of degrees of freedom left, and might conceivably have kinetic energy although the containing surfaces and immersed bodies were all at rest, yet we may suppose the relative positions of all the particles of the fluid to be determined by certain generalised coordinates ^1...$'^, t' being sufficiently great, and ^^^^ • • • $'n being the remaining coordinates of the system, those namely which define the position of the containing 26.] INCOMPRESSIBLE FLUID. 45 surfaces and immersed bodies. If now the containing surfaces or immersed bodies be set in motion by any impulses from rest, we have already seen that the kinetic energy of the whole system is the least which is consistent with the velocities assumed by those surfaces and bodies, that is with q^^^ ... ^„, and therefore, by Art. 20, the generalised components of momentum corre- sponding to qi •.. qr ^^^ severally zero. It is evident also that any impulses which if applied to the system at rest would make j)i ...j)^ zero, will not if applied to the system in motion, how- ever they may alter the velocities, give to pj^ ...p^ any values. Hence, so far as impulses applied to the surfaces and immersed bodies are concerned, p^ -" Pr remain zero for all time. If any finite forces act on the system, the same result as re- gards jo^ ...pj. follows from Lagrange's equations. For dt dq dq for each of the coordinates ^i-..^^* Now if the containing surfaces and immersed bodies were all at rest and fixed in space the forces acting on the system could have no tendency to pro- duce relative motion among the particles of the fluid, it being homogeneous. Hence for q^,.. q^, -7- = ; and evidently also -^ = 0. Hence -f- = 0; and the motion bein^ from rest, » = 0. aq dt ' o jj It follows that in such a system as we have supposed, to whatever finite or impulsive forces it may be subject, provided the impulses act at points in the containing surfaces or immersed bodies, the components of momentum jOj ...pr are always zero. And therefore the kinetic energy of the entire system at any instant can be expressed in terms of the momenta corresponding to the remaining coordinates ^^+j ... ^„, which define the posi- tions of the containing surfaces and immersed bodies. It follows, by Arts. 20 and 25, that the motion of the entire system at any instant is that which it would take if, the whole being at rest, their actual velocities at that instant were im- pulsively given to the containing surfaces and immersed bodies. If, therefore, the positions of these containing or immersed 46 KINETICS OF [27. surfaces are determined by a certain finite number of coordinates, the whole motion of the fluid and of the immersed bodies may also be determined in terms of these coordinates. 27.] The first two of the following three examples are taken from Thomson and Tait, p. 262, &c. Exam.^le 1. A ball is set in motion through a mass of friction- less incompressible fluid extending infinitely in all directions on one side of an infinite plane and originally at rest. The position of the ball, and therefore by our general pro- position^ the whole motion is determined if the coordinates of the ball's centre ^, y, z at the time t are known. Let the axis of x be taken perpendicular to the bounding plane through any point whatever of that plane ; then the kinetic energy T must be a quadratic function of x, j?, and z, with co- efficients certain functions of x, y^ and z. It is clear that T remains of the same value when either i/ or z has its sign reversed, and therefore the terms in xy^ xz, and yz do not occur in T, which is therefore reduced to the form where P, Q, and B are functions of x only. From the symmetry it is clear that Q = R, and hence T=i{Fx' + Q{f-\-z')}. If therefore X, Y, and Z are the generalised components of force corresponding to x, y, and z, Lagrange's equations give us Examjple 2. A solid of revolution moving through a friction- less incompressible fluid infinitely extended so as to keep its axis always in one plane. In this case there are three degrees of freedom, and therefore three independent velocities in terms of which the whole motion may be determined. Let these be chosen as the two components of the velocity of any point in the axis of figure, and the angular velocity about 2 7-] INCOMPRESSIBLE FLUID. 47 an axis through the same point perpendicular to the plane in which that axis moves. It is assumed that the body has no rotation about its axis of figure. If % and c[ be the resolved parts of the velocity of the point along and perpendicular to the axis of figure, and w the angular velocity about the axis through this point, it is clear from our general proposition that the whole kinetic energy of the body and fluid is For the reversal of the sign of u cannot affect T. A, B, C, D also are obviously constants, since the liquid is of infinite extent. By properly selecting the aforesaid point in the axis the equation for T may be reduced by obvious reductions to u and V being the velocities of the new point in the aforesaid directions. If 6 be the angle between the axis of figure and the axis of a?, and X and ^ the coordinates of the aforesaid point, we get 10=^6, u =^ xcoBO + ysinO, v =i y cosO—xsinO ; Au sin 6 -\- Bv cos 6 ; dT ,. „, dT ^ dT ^ Also, if A, f, T} be the generalised components of impulse cor- responding to w, Xf and ^, A = — : = IJ-d, dd ^ = AucosO — BvsinO, 7} = AusinQ + Bvcosd. And Lagrange's equations give us, ^S + W {{e-r}')sin2e-2.ivcosd} = Z, ^_ T ^- y. dt~ ' dt~ ' dT — , i dT dT de ■ Ed, dx ~ A ucosO- -BvBind, dy 48 KINETICS OF [27. i/, X, and Y being- generalised components of force corresponding to Wj cc, and y. If i, X, and r each = 0, and if the axes be so taken that r; = 0, as is clearly possible in this case, we get The case of the common pendulum where (p = -- Example 3. As a third illustration we may take a case of motion in three dimensions as follows. An ellipsoid of revolution moving in an infinite mass of friction- less incompressible fluid — no forces. If u^ V, w be the velocities of the centre resolved parallel to the three principal axes, and if coj, (Hc^^ 0)3 be the angular velo- cities about these axes, it is clear from our general proposition that the whole kinetic energy, T, of the liquid and ellipsoid may be expressed as a quadratic function of these six quantities ^, ^, w^ <^\i ^2) ^3- Also from the perfect symmetry it is clear that terms involving the products of these quantities cannot appear in T^ and there- fore that where A^ J?, and D will be certain constant quantities, and where C is the moment of inertia of the ellipsoid about its axis. If a?, y, z be the rectangular coordinates of the centre, and if 6y (f), yj/ be the angular coordinates of ordinary use in determining the orientation of a rigid body, we get by obvious substitution and reduction, dT -jr = Au {cos \// cos (^ cos ^ — sine/) sin \//^} — il V {cos \/a sin (j^ cos + cos (/) sin \//} + Bw cos y\f sin 6, dT -jr = -4 w {sin \|^ cos ^ cos + cos y\r sin ^} -\-Av {cos(/)Cos\/a — sin\//sin(/)cos^} + ^w sin \/^ sin 0, dT -— - = Au sin 6 cos w; ; dT and substituting in -^ = F, we get w/ = ^ cos ; whence substituting in the equations, giving x^ y^ z in terms of u^ Vy and w^ we get x=. F{— — — ) cos ^ sin ^ cos \/r, y zn F (- — ) sin ^ cos sin \/r, A. Jj A Jj Since CO3 = <^ + \^ cos ^, 0)2 = ^ cos (^—\^ sin ^ sin <^, oji = ^ sin (^ + 1^ sin ^ cos <^. It follows that T is independent of i//^, and therefore Lagrange's equation corresponding to the coordinate ^Ir becomes d ,dT, ^ dT ^ , ^ ^. y- ( — r) = 0, or ---r = E (a constant). "^ dyjf dyjr But -^ = I)(o^—4 +i)a)2-^ +C(03 — ;^ cZa/t dyjr dyj/ dyfr = Z>sind {a)iC0S<^— toasint^} + (7a>3C0S^ = Z^sin'^^A^+C'cOgCOS^; .*. 2)sin2^\/r + (7(«)3Cos^ = ^. And T may be reduced to <-!•, °'^-*^ '*'/^^>*> 50 STATICAL ANALOGY. [28. 28.] Lord Rayleigh has pointed out a remarkable analogy between the dynamical theorems hitherto demonstrated and certain statical theorems, generalised components of velocity being replaced by small displacements, generalised components of momentum by impressed forces, and kinetic energy by po- tential energy of deformation. For example, suppose a statical system under the influence of given impressed forces, which are either constant or functions of the positions of the particles, to be in a position of stable equilibrium ; the potential energy of the system must then be a minimum. Let it be f^ . Let the system be slightly displaced by the application of certain additional forces, and let the gene- ralised coordinates of the displaced position reckoned from the position of stable equilibrium be ^^ ... ^^. The potential energy of the displaced system will then be Vq + V, where F is a quad- ratic function of q-^ ... ^„, involving generally coefficients func- tions of the coordinates, «, ^, c, . . . , of the position of stable equi- librium. Fis defined to be i:hQ potential energy of deformation. Also the generalised components of force P^... P^, required to produce the deformation q^-'-^.n') ^^^ linear functions of ^^ . . . §-„ with coefficients functions of a^h^ c^ ,.. such that We can then by means of these linear equations prove a series of propositions exactly analogous to Propositions IV-VII of Art. 9 ; and in particular we can prove that if Pj . . . P^ be the forces producing the deformation $'1 ... $'„^ while P/... P/ produce q{. . . q^ from the same position of stable equilibrium, then ^F4-^r,, or 2^/=25:,. Then we may prove a proposition analogous to that of the maximum kinetic energy (Art. 1 6) above, namely, that if such a material system be held in equilibrium^ in any position slightly displaced from that of stable equilibrium, by means of forces applied from without, the potential energy of such displacement will be greater the greater the number of degrees of freedom, 28.] STATICAL ANALOGY. 51 and that if the system be subject to any constraints, and so con- strained be held in equilibrium in a position slightly displaced from the original position of stable equilibrium by means of the same external forces as before, the potential energy of the free system in its displaced position will be greater than that of the constrained system in its displaced position by the potential energy of the difference of displacements in the two displaced positions. Let qi ... q^he the displacements in the displaced position of the free system,, reckoned from the position of stable equilibrium, and let ^^ be the potential energy of displacement in this case. Let q^.,. q^ and V^ be. the corresponding quantities in the con- strained system. Let P^ ... P^ be the external or additional impressed forces in both cases. Then we have, as above stated, and A = ^' ^2=P-. &c. In the constrained system it will no longer be true that dV^ dj^ ^~ dq(' ^^ dq;'^^'' because the displacements are no longer independent, but, by reasoning in all respects analogous to that of Art. 16 above, we must have, by the principle of virtual velocities, - >(?-f)^- • (^) Also in the constrained system therefore F. = 4.f/; = isf (.-/)• E % 52 STATICAL ANALOGY. [29- = Vq_g'. (See Prop. VII. Art. 9.) That is, F-K/=K ^4-9'' whence the proposition is proved. 29.] Again, if the expression 2PQ~/q be denoted by 4^, where the P's are the given forces, and the Q's any whatever small displacements, and /q the potential energy of deformation corresponding to the Q's, and if xjr and xj/^ be the values of 4^ when q and / respectively are substituted for Q, that is \//- = ^^, \/a'= ?J', then we may easily prove, as in Art. 18 above, mutatis mutandis J that from which follows, as a particular case, the result already obtained, F,- r, = ^-^'= i 2 . (^ - ^) to-/) = r,_^. 30.] Hence we may shew that if the stiffness in any part or parts of the system be diminished, the connexions remaining unchanged, the potential energy of deformation will be increased. For if the displacements were the same it is evident that the potential energy would be diminished, there being less stiffness, that is, ^q < ^5 if ^q be the potential energy of deformation in the new, F^ in the original system with the same displacements Now in the function 4^_, formed for the new system, let the Q's be the original q's, and let *' be the value of yj/ in this case. Then *'=2P^-F/; .-. ^' > 2 Pq-Vq because F/< V^ ; .'. ^'>F„ since 2Pq-V^=r,. But if qi- . . qn be the actual values of the displacements in the position of equilibrium of the new system under the impressed forces P, it follows, as above proved, that F/ > 4'' ; therefore, a fortiori f F/ >'P^, 32.] STATICAL ANALOGY. 53 31.] There is also a statical analogue to the theorem of mini- mum kinetic energy of Art. 20, which maybe stated as follows : — If a material system be held in a deformed position with given values of certain of the displacements, suppose ^i ... ^r> reckoned from the position of stable equilibrium, then the po- tential energy of deformation will be the least possible when the external or additional forces by which the displacements are produced are exclusively of the types corresponding to those displacements, and the potential energy of any other deformed position having the same values of qx--- 9.r exceeds this least potential energy by the potential energy of the displacement which is the difference of the two positions. The proof of this is analogous to that of the corresponding dynamical theorem. Let P-j^... P^ be the forces necessary to produce the given displacements ^i ... ^^ when P^+j ... P„ are severally zero. Let V be the potential energy of deformation in this case, F^ that in some other deformed position having the same values of q^ >^^qr\ and let P^-\-P(^ P^-\-P^^ &c. be the forces, and q^ + q{^ q^ + q^^ &c. the displacements in the latter case ; then by hypothesis every ^ from q{ to q^ inclusive is zero, and every P from P^+i to P„ inclusive is zero ; therefore 2P/=2P'g = 0; therefore F'= i 2 (P + P') (g + /) = i2P^ + i2P'/ = F+JSP'2', as was to be proved. Hence we can deduce a theorem corresponding to that of Art. 2 1 , viz. If a material system in stable equilibrium under the action of its own forces undergo any small displacement or deformation by fresh forces applied from without, being so forced into a new position of equilibrium, the potential energy gained by such deformation is the least which the system can have consistently with the displacements, whatever they may be, of the pointa at which the fresh forces are applied. 33.] Our dynamical equations have also analogues in electro- statics. It can be shewn, for instance, that in any system of conductors in equilibrium relations exist analogous to those 54- ELECTROSTATICAL ANALOGY. [32. established for a dynamical system, generalised components of momentum and velocity being replaced by the potentials and charges of the several conductors, and kinetic energy by the intrinsic energy of the system, that is to say, the whole work which would have to be done to bring the charges from an infinite distance to the several conductors against their mutual repulsions. It is understood that the charges of the same sign repel one another according to the law of the inverse square. Let C-^,., Cghe the several conductors, qi,..qn the generalised coordinates defining their positions in spaccj ^1 . . . ^^ their charges, 7^1 .. . T^ their potentials when the system is in equilibrium^ and E the intrinsic energy. Then the work which would have to be done to bring an infinitely small quantity of electricity, dcj to the conductor C^ from an infinite distance is evidently T^de, Hence we obtain generally de Again, let us suppose all the charges to be originally zero, and to be gradually increased pari passu in the same ratio till they attain their value in the actual system ; the potentials at any instant during this gradual variation are proportional to the charges at the instant. It follows, as shewn by Maxwell, Elec- tricity and Magnetism^ part I, chapter iii, that each potential is a linear function of all the charges, with coefficients depend- ing on the forms of the conductors,, and the coordinates q^,., q^ defining their positions in space. It follows also that E = J 2 Fe ; * and E is therefore a quadratic function of the charges having coefficients functions of the ^^s. If the charges e^ ...e^ produce potentials f^.-.f^} while e-^. . . e^ produce V{...V^\ evidently e-^^e{, &c. will produce ?J— ^', &c. * For suppose the charges to be introduced uniformly during any time. Then after time t they will be Kit ...Egt, and the potentials Cit ...Cgt, where the K'a and C's are constants. Then, since -7- = V, we have de 8E = SF8e = :SCEt8t; whence E = ^SCKt^ = h'^Ve, no constant of integration being required, because when « = E = 0. 33-] ELECTROSTATICAL ANALOGY. 55 If the linear equations be JJP then, since -7- = F^ we must have J^g = -^21^ ^^j ^^^^ i^> generally, dF^ __ d;?/^ Conversely, every e is a linear function of all the Ps, and E may be expressed as a quadratic function of the F's with coeflScients functions of the ^^s. When so expressed we shall write it Ey, and when expressed as a quadratic function of the charges, Eg. It follows then from the linear equations connecting F and e, that dEv Also that if ^^ . . . ?^ be the potentials of the several conductors when the charges are e■^^,..eg, and if F^. ..F^' he the potentials of the same system of conductors in the same positions when the charges are e/... e/, then 2Fe'=2F'e. This result can be established by an independent method ; see an article by Clausius in the Pkilosoj)Mcal Magazine^ vol. iv. Fifth Series, p. 454. 33.] We can now prove as in Proposition III of Art. 9 that dEe_^ dEv ^ ^ dq dq the potentials in the one coefficient, and the charges in the other, being treated as constant, and the forms of the conductors in either case unaltered. For since E, + Ep = 2 E = 2F«, let us suppose e^ F, and q all to vary. ^ Then we have the summations being for all 'bq's or all c)e's &c. as the case may be. 56 ELECTEOSTATICAL ANALOGY. [34. But -— ? = V and —-^ = e; de dV therefore the above equation is reduced to in which the summation is for all the ^^'s. And since the ^ is the diminution of the intrinsic energy of the system consequent on the conductors undergoing the displace- ment in space denoted by <)^, all the charges remaining unaltered ; and therefore =-^ measures the mechanical force dg tending to displace the conductors in the manner denoted by t) q. Similarly ~- is the mechanical force tending to displace them in the same manner, if by any means the potentials be maintained constant during the displacement, while the charges vary. And the equation just obtained shows that the resultant mechanical forces are equal and opposite in the two cases. This result is obtained in a different way by Maxwell in the work above referred to, vol. i. p. 95. 34.] If any two or more conductors originally insulated be connected together, so as to form one conductor, they acquire of course uniform potential, and a new distribution of their charges takes place, the potentials of other parts of the system undergoing corresponding alterations. If 7J . . . ?; and ^^ . . . ^^ be the original potentials and charges, and T[\.. ¥^ and e{ ... e^ those after the connexion is established, we can prove the following theorem, viz. SFe' = 2 Y'e = 2FV or 2 (F- F') /= being the analogue of the equation 2(p-/)2' = 0, deduced in Art. 16 from D'Alembert's principle. 34-] ELECTROSTATICAL ANALOGY. * 57 For in the case of every conductor which retains its insulation e = e\ and therefore P^= V'e\ In case of a group of conductors which become connected, V is the same for all members of the group, and the sum of the charges is unaltered. Therefore 2 denoting summation for the group. It follows that for the entire system 2Fe'= SF'e = 2FV (1) Hence we can prove a theorem analogous to that of Art. 16, viz. that if any two or more conductors be connected so as to form one conductor, the intrinsic energy of the entire system is diminished by an amount equal to the intrinsic energy which the system would have, if the charge on each conductor in the entire system were the difference between its charges in the original and altered state ; that is, that 2Fe = 2FV + 2(F-F0(6-O- For -^sVe-^Y'e' = ^Ve-^Ve\ = 2F(e-0, = 2(F-r)(6-0; because 2F'(e-e') = by (1); i.e. 2Fe = 2FV + 2(F-F')(e-0. as was to be proved. The loss of intrinsic energy is therefore equal to the work which would have to be done to bring to all the conductors, supposed originally uncharged and insulated, the charges e^e\ It follows that if a given quantity of electricity be distributed over a surface, the intrinsic energy is the least possible when the distribution is such as to make the potential uniform over the surface. And the same law holds for a number of surfaces if the charges on each be given. Hence also if a number of insulated conductors be so charged as to have potentials J^... J^ respectively, then if they be all connected together^ they will assume the common potential where a is for each conductor proportional to the charge which the conductor has after the connexion is established. 58 ELECTROKINETICS. [35. 35.] Again, in electrokinetics the principle of minimum kinetic energy can be applied to establish the theory of induction currents. For instance^ let there be a number of wires C^.^.C^ each forming a closed curve or circuit. Let electric currents be set up in these wires. If we denote by ^^ the quantity of electricity that has passed in the positive direction through a section of the wire C^ since a given epoch, the current in the wire C^ at any instant will be represented by -~ or <^j . With this notation the electrokinetic energy of the system at any instant is See Maxwell's Electricity, Vol. II. Art. 578. In this expression the coefficient i/^ is // cose r 1 where ds and ds are two elements of the first circuit, - the r mean inverse distance between them"^, c the angle between their directions both taken the same way round the circuity and the integration includes every pair of such elements. L^..,Ln have corresponding values for the other circuits. In like manner the coefficient if^g i^ !S~ where ds-^ is an element of the first, and ds^ of the second circuit, r and e having the same meanings as before. In the language of quaternions, if p^ be the mean vector potential of the first circuit, pg that of the second^ and so on, * The wire having small finite thickness, let a be the distance from a point in a section of ds to a point in a section of ds : then - is the mean value of - for all r a such pairs of points. In forming the mean vector potential for the circuit - has the same signification. 35-] INDUCTION CURRENTS. 69 M,^=--j- ISp^ds^=- Y-hpi^hr &c. = &c. Now, there being initially no currents in the wires, let a current (^^ be suddenly generated in the wire C^ by an electro- motive force applied to that wire, all the wires remaining at rest in space. It is then observed that currents make their appearance simultaneously in the other wires. These are called induced currents. Their values at the instant of the current ^^ being created, that is before they sensibly decay by the resistance of the wires, are determined by the condition that the electrokinetic energy of the system is to be a minimum consistently with the existence of the current ^^ in the wire C-^ . That is by the equations — -0 — -0 in all as many equations as there are induction currents to be determined. To take for simplicity the case of two circuits, if the current <^i be suddenly generated in C^ by an electromotive force applied to (7i, then in order to determine the current ^^ induced in C^, we shall have the equation dT — ;- = or if(^i + Z2<^2 = 0; that IS, 92 = F" 9i- ^2 Now Z/2 is a necessarily positive quantity. Therefore <^2 ^^ in the same direction round the circuit as (^j, or in the opposite direction, according as M is negative or positive. If for instance both wires be circular and in parallel planes, and so placed that the projection of the first on the plane of the second lies outside the second, M will be negative, and ^.^ therefore in the same direction as ^^. That is, both currents viewed from above may be in the same direction as the motion 60 INDUCTION CURRENTS. [35. of the hands of a watch. It follows that if we compare those portions of the two wires which are nearest to each other, the current in C^ will be in the opposite direction in space to that in Cj. If one be from south to north, the other will be from north to south. This agrees with the observed phenomena. See Maxwell, Art. 530. If the current ^^ be generated gradually, the rate at which the current ^2 ^^ destroyed by the resistance of the wire will generally bear a finite ratio to the rate at which it is generated by induction. But if the resistance be very small, we shall have the equation dt L^ dt to express the rate of variation of ^2 ^^ terms of that of ^ T dd>^ ; dM whence dipi _ dM M^^ — L^ <^2 ~dt~~dt' RL..-M^ ' 'l-"2 d^^ _ dM M^.^—L^ <^j 'dT ~~dt" Ly^L^^—M^ 35-1 INDUOTION CURRENTS. 61 If q be one of the coordinates defining the position of the second wire relatively to the first, we shall have evidently for any displacement denoted by c)^ d^y^ _ dM M^^ — L^^^ dq ~~ dq LyL^ — M^ di^2- dq dx _dM 5 r <^4 , ; <^<^i? ^ dM^ 2M^^^^-L,^^^-L^4), dq' L^L^-M^ Now L^, L^, and L^L^-M^ are all necessarily positive, and therefore a/L^ i/g — if is necessarily positive, and 2M^,^,-L,,'-L,4>,' is necessarily negative. Hence the last equation shews that ~J^ is necessarily negative. That is, the effect of displacing the second wire in any direction relatively to the first is to generate in the two wires induced currents which diminish the force tending to cause displacement in that direction. This agrees with the observed phenomena. See Maxwell, Art. 530. CHAPTEE III. CHARACTERISTIC AND PRINCIPAL FUNCTIONS. Article 36.] Definition. If T be the kinetic energy of dmj material system, and if A be equal to the definite integral Tdt, '-f -JtQ A is called the Action of the system from the time tQ to the time t. In any conservative system^ the Action between two given positions of the system may always he exjpressed in terms of the initial and final coordinates of the system and the total energy^ and when thus expressed it satisfies the equations of which the following are types : dA dA „ dA Tg=^- i^.= -^»'*<=-' SE = '-*«' {9j P) i^^'^9 <^'^y coordinate and corresponding momentum in the final position^ and {%^ p^j leing the values of these magnitudes in the initial position, and E being the total energy. For if the time t be reckoned from the beginning of the interval, the g'^s and the jt?'s may by proper equations be ex- pressed in terms of the q^% the initial values of the momenta, and the time t, and therefore T may be similarly expressed. Also, if TJ be the force function, and E the total energy, we know that ^ — ZJ = E . By means of this equation and those last referred to we can eliminate t and express the initial momenta in terms of the ^o's, the 3''s, and E f. * That is, a system in which the forces possess a force function. + It is important to observe that the process in the text will give generally more than one set of initial momenta with which the system can pass from the given initial to the given final configuration. To each set corresponds a distinct dA type or value of .4, and a distinct set of final momenta. The equations -r— = 'p, &c. hold for each type or value of A in relation to the corresponding momenta. CHARACTERISTIC FUNCTION. 63 Hence T, and therefore A^ may be similarly expressed in terms of the ^^'s, the ^'s, and E, and therefore the first part of the proposition is proved. Affain, since rt .*. A = S mv^dt = S mvds = 2/ pdq by definition, JtQ JtQ Jqq where S denotes summation for all the elements of mass. Let any possible variations be given to all the variables, then JqQ JqQ Integrate the first term by parts and we get hA =2U?8^ ^0 2 Ubpdq — djphq) where But = jf>bq-p^bq^ a^ = 2 ^^-di + 2 Qq ^£(ibp-pq)dt. dp_^ dT^^ and -f =z Fa j— , dt ^ dq ' ,. bA = :2{pbq) +2 ('^bp + ^bq^F,bq)dt, =.2{pbq) + I bTdt-2 I Fqbqdt, /7TT But in this case i^^ = -7- , the system being conservative ; .-. 2rF^bqdt= pbUdt; • f\bT-bU) JtQ bAzn'S.Ubq] + dt = 2 = 2 pbq\ + / 5E dA dq = P> pbq\ +(<-^^)5E; dA dA ^ ^ 64 JACOBl's EQUATIO:tTS. [7,y. When the Action A is thus expressed as a function of the initial and final coordinates and the energy it is written y (^0 • ••$'••• E), or more briefly y, and is called the cJiaracteristic function. It is clear that y satisfies a partial dijfferential equa- tion in the n variables ^i, q^^-.q^n of the first order and the second degree, namely the equation which results from writing J- > -^ > &c. for jt?!, jp^y &c. in the equation dq^ aq^ ^^= tr+E. 37.] We next prove the converse of the last proposition : If the partial differential equation in f he formed hy writing y-i -j-'i 8fc. for pi^ jo^i ^^' ^^ ^^^ equation of conservation of aqi dq^ energy of any conservative system^ and iff he any solution of that equation^ then an actual motion of the system may he determined hy making j^i = ;j-» j^2 = 7 ' ^^'> ^^^^^^ i^i5 Aj ^^' '^^^ ^^^ gene- ralised components of momentum. Since f satisfies the partial differential equation formed by substituting -j-^ j-^ &c. for p^, p^^ &c. in the equation therefore f must satisfy the n equations of the type dT dT.dp dT,dp^_dU_ dqr dp^ dqr dp^ dq^ dq,. when -^ J -r- &c. have been written for p^^p^s &c. in 21. dq^ dq^ ri /2 p But by Proposition (III) Therefore /satisfies the equation dT^ . dj . (Zy _1?_2^. ■ dqr dq^dqr dq^dq^ ' dq^ ^* dTj, ,. d . d ^ .df 38.] JACOBI'S EQUATIONS. 65 dqr dt dq^ *" ' if jOy be taken = -— ; that is, the motion determined by making p —~- satisfies Lagrange's equations, and is therefore a natural motion of the system. 38.] If a complete primitive of the partial differential equation referred to in the last article he found in the form /(5'i---5'««i •••««-!) + ««. where a^, a^.,.a^ are any arbitrary constants^ then the integrals of the dynamical eqtiations will he where /3i ... /3„_i E are n additional arbitrary constants. -For since f satisfies the partial differential equation T,= U+E,, (1) when jt?i, J925 ^^- ^^^® ^^^^ replaced by -=— > -y-> &c., it follows dq^ dq^ that f must satisfy the n — \ equations of the type dT^ dp^ dT^ dpn_Q dp^ da^ '" dpn ^«i * with the additional equation d^dp^ d_T^dp^_ dp.'dE"^ '" "^ dp^ dE found by differentiating (l) with regard to a^, ...a„_i and E successively. dT Now, with the substitutions referred to, -,— ^ = qi, &c., these equations become (2) &c. from which q^y q^^j... in can be determined. F 66 PRINCIPAL FUNCTION. [39. But if we differentiate with regard to t the n equations we obtain precisely these last equations (2) to determine the magnitudes q^, q^... qn- For instance, differentiating ~- = /3, with regard to t we obtain d df _ , . d . d . df _ did^,~^'^'d^,'^""^ ^" diJ d^, ~ ^• Whence the proposition is established. 39.] Definition. If A represent the Action in any conservative system where the time does not enter explicitly into the con- necting equations, and if S be determined as a function of the initial and final coordinates and the time, by means of the equation the function thus found is called the Principal Function ^. If B he the principal function in any conservative system where the time t does not enter explicitly into the connecting equations^ then dS dS dS where q^ represents any one of the initial, and q any one of the final coordinates. For since S-= A — 'Eit, .-. hS=hA-Eht-thE; and therefore if the final coordinates alone vary, dq dt dq ^ dE dA , dA But — - = « and -~- = t, dq ^ dE ' " dq ~^' dt ~ * The time is so very generally reckoned from the beginning of the motion that unless the contrary be expressly mentioned it will be assumed that t<, = 0. 41.] CHARACTEKISTIC AND PRINCIPAL FUNCTIONS. 67 and by varying the initial coordinates we obtain similarly, dS ¥o = -^»- 40.] Since ^ is a function of the initial and final coordinates, represented typically by q^ and q, and of E, the increase of A per unit of time as the system passes through the configuration q is clearly dA ^ .dA dt ^ dq We may conceive the system passing through the same con- figuration q with any other velocities q and the same value of E. The increase of A^ considered as a function of q^ and q, per unit of time in this latter motion is ^ .,dA Now, since the kinetic energy in the configuration q is the same for both motions and dA it follows from Art. 1 that the increase of A per unit of time is greater in the actual motion than in the / motion, and, given E, is a maximum in the actual motion. This is true for every material system. In the case of a free particle of mass m we have p = mg, ^ij $'25 % being the rectangular coordinates of the particle, and the equation p •=. -y in this case expresses the fact that the path of the particle is normal to the surfaces of equal action. By extending the mean- ing of the terms ' normal ' and ' surface,' we might say generally that a motion in which p is proportional to ^ is normal to the surfacey= constant. If t instead of E be invariable, similar statements apply to the principal function 8, 41.] As an illustration of the formation of the principal and F 2 68 CHARACTERISTIC AND PRINCIPAL FUNCTIONS. [4 1. characteristic functions, let us consider the case of a projectile of mass unity. Let the point of projection be the origin of co- ordinates, X andy the horizontal and vertical coordinates of the projectile. Let the initial and terminal horizontal and vertical velocities be Uq^ Vq, u, v. Let the time t be measured from the instant of projection. Let the potential at the point of pro- jection be zero. Here we have u' + v'' + 2gy = V + t'o' = 2E; u=zu^, v=:v^-gt; gf gf X = u^t, y — v^t-~=vt-\-—\ (a) .-. E=i,j.^ + (,+ fr}; (/3) t'' = \ {2E-^2/ ± y4E2-4E^2/-<7'^'^'}. (5) S=A-E. = t^-^-^. The expression for S on the right-hand side of 8 is the Prin- cipal Function, and on being differentiated with regard to a?, y, and t respectively, attention being paid to (a) we shall obtain the quantities u, v^ and — E. The expression for A on the right-hand side of (y) is not in its present form the Characteristic Function, but we may obtain that function by substituting in (y) the value of t obtained from {^). Thus let A in (y) be differentiated with regard to ^, x and E being constant, and we get 42.] CHARACTERISTIC AND PRINCIPAL FUNCTIONS. 69 Also from (/3) dt 1 _ 1 2 E t — = — + the + sign being used if the — sign be used in the expression for t^j and vice versa. Also from (^), ± s/4E'-4Egy-g'a^' = ^ +gy2E, = v,gi^2E. Hence we obtain dt , ^^, v.gt — 2E 2E -j-(v.qt-2E) = ^-^ ■ — , dy^'^ ^ gt gt = -Vo; dA 2y and -;— = — ^ —Vf, = V. dy t By the same process we may obtain dA_ dx If in the above formulae we were to write x^Xq, y—y^i ^^^ X and y, taking ^q, y^ for the initial coordinates, we might obtain by the same method dj^_ _ d^ __ dx- ""^ dy- ^- 42.] As another example of the formation and properties of the Characteristic and Principal Functions we may take the case of the elliptic orbit under a central force \i.r. The equations of motion are in this case d'^x ^ /-x ^ + '^"=<»; <') S+''^=« (2) 70 CHAEACTERISTIC AND PEINCIPAL FUNCTIONS. [42. The integmls are a? = a cos V fit + b sin VfJit, y = a' cos '//x^ + fe'sin '/jJi^; dx _ "jT — v/x{6cos v/x^ — (xsin v/xi}, c?2/ _ ■^ — v/x{6'cos v/x^ — a'sin v/x<}. Whence we easily get (remembering that the force function U is ^ -r4:)^^-r(i)^<^' = fx [ — - — t H — { — - — sm 2 Vi^t — ab (1 —cos 2 v fx<)}j 2 2v/x '^ + /u[ 1+ — ^ { sin2\//u^-a'6'(l-cos2A//xi)}]; ^ 2 V jix '^ and =:^ | <^' + 6^^-«^-a^^ ^.^ 2 ^/;i^_(a6 + a'60(l -cos2 //lO} • 2 A But if ^0, ^Q be the initial coordinates, we have ^0 = «> 2/0 = «' ; a? — a^ocos \//x^ , y — y^cos ViJLt = : — T= ' ^ = : — -7= ; sm V/x< sm vfxi whence by substitution S is easily reduced to ~Y- {(^ + / + 'V + 2/o^) cot Vixt-2{xx^-^yy,) cosec a/^^}- And thus the principal function is found. It will be seen also that S satisfies the two difierential equations 42.] CHAKACTERISTIC AND PRINCIPAL FUNCTIONS. 71 To find A we must first of all determine ^ as a function oi Xq, ^q, X, y, and E, and then eliminate t from the expression for S. Now — = a''-\-a'^ + b^ + b"' _ (a^-a^, cos Vfi tf + {y-y, cos a//x ty — ^0 -r 2/0 "1 . o /-, » sm^ V /ut < __ a?2 + y' + < + 2/o^-2(a7a;o + yyo)cos>//x< ^ sin^ \/ju t .-. COSA/fX^=^{a7a?o + 2/2/o 2E ± V 1- ^(^' + 2/' + < + 2/o^) +^1-2(^^0 + 2/2/0)'. If the value of t thus found be substituted in S and E ^ be added, we obtain the characteristic function J, and it will be found that A satisfies the equations dA dx dA dx dA dy, dA _ dy ^ dx ~~ dt dxQ dt^ dy dt dy^ dt^ and the partial differential equations "0 CHAPTEE IV. STATIONARY AND LEAST ACTION. Article 43.] Let a material system be in motion under the action of any conservative forces, and in tlie interval between the times t^ and t let it pass from any given configuration to any other. Let A be the action between these two configurations so that A = 2 mv^dt, JtQ and let T be the kinetic energy, U the force function, and E the total energy at any instant during the motion. Let the motion of the system be ideally varied^ so that while the initial and final configurations remain the same as before the system shall pass from one to the other through a series of configurations always indefinitely near to some configuration in the actual motion, and also so that the equation T-U = E remains true for the same value of E throughout the varied motion. Such a varied motion is ideally possible^ but can in general only be effected actually by the introduction of ad- ditional constraints from without. Then, in such a case, the small variation bAin the value of the action in passing from the original unconstrained to the varied constrained motion is al- ways zero. This is the principle of Stationary Action, For A = ^1 mv^dt = ^ Tpbq; therefore, as in Art. 36, hA z=2{pbq) ^2r{qbp^^bq)dt go JtQ dt for any small variations whatever 6j^? and 6^'. EXAMPLE ILLUSTKATIVE OP LEAST ACTION. 73 And this equation reduces, as in the article mentioned, to bA = 2(2>bq) + (\hT-W)dt. But the initial and final coordinates q^ and q remain un- altered, as also does jP— TJ by our hypothesis, .-. {^jphq) =:0 and hT-bU = Q', .-. hA = 0. 44.] Exactly in the same way it may be shewn that if the time be the same in the two courses, but E vary, then when the initial and final coordinates remain unaltered, 8x9 = 0. For 8iS'= 8^-(«-08E = hA-{t-t,) {hT^bU}, = ^{pbq) -^ f\bT-bU)dt-{t-Q(bT-bU), = 0. 45.] In the above expressions bA and 8 /S respectively include the first powers only of small variations according to the ordinary notation of the Differential Calculus and the Calculus of Varia- tions ; and the Principle of Stationary Action just proved shews that the difference between the actions in the original and varied motion is to the first approximation zero; A therefore satisfies the first condition of being a minimum. We proceed now to investigate the sign of this difference when higher orders of the variations are considered, and the final result will be to shew that when certain conditions are fulfilled the Action will be a true minimum, and that when these conditions are not fulfilled, no general rule can be asserted concerning it. Before treating the general question we will consider the simple case of the projectile. For the sake of brevity we will again suppose the origin to be the point of projection ; then as before < + ^,^ or F2 = 2E, where F is the velocity of projection. Let also a be the angle of projection so that i(q = V cos a, Vq'^V sin a. . ^ 74 EXAMPLE OP LEAST ACTION. [45. Then, if t be the time in passing to the point x^ y, t'--^{y'-9y)±^^fy'-'^y'9y-t^'\ (i) and if a be the angle of projection, in order that the projectile may pass through ^, y, ^^^^ri±^yrEW^^7EIl. (11) gx From (II) we obtain V sin a and V cos a, or the initial values of the momenta in terms of F, x, y. Again, as we found above, ^ = ?!+l! + Z«a (HI) and if we substitute for t in this expression the value obtained in (I) we obtain the characteristic function f^ or the expression for A in terms of a?, y, and V^ (i. e. 2E). From (I) we see that ^=± /\/-~ (V'-gy) ± jVV^-2V^9y-fay^. And as a negative value of t has no meaning bearing upon our question, we shall reject it and take t = /\/-^ ( V'-gy) ± VV'-2V'gy-g''x\ . Similarly, in determining Fsina and Fcosa, we shall take for sin a and cos a, in terms of tan a, the values H - and H respectively. VI + tan^ a V 1 + tan^ a Whence it appears that when the time and the two initial momenta are determined in terms of x, y, and F^, each of the^e quantities will be expressed by two distinct functions of Xj y, and F^, owing to the double sign of the radical V'V*-2V^gy-g''x^; and the same may be said of A, since by (III) there corresponds a value of A to each value of t. 45.] EXAMPLE OF LEAST ACTION. 75 Fig. 2. Hence we learn that if a particle be projected from with a given velocity, and be required to pass through the point P whose coordinates referred to are x and y, the necessary- horizontal and vertical momenta at and the action from to P will be given in each case by either of two distinct functions of a?, y, and V^j so that in general there are two distinct courses from to P, viz. OC^P and OC^P^ having different times of passage, different initial mo- menta, and different values of the Action. If however the point P be so situated that then the radical vanishes, the two functions mentioned above coincide in value in the expressions for momenta, time, and Action, and the two courses from to P become coincident in all respects. The locus of P thus determined is clearly the parabola AGC\ touching the common direc- trix of all the parabolas at the point A, vertically above the point of projection 0, and having its focus at 0. If we find the envelope of the curves y = irtan a — — -=v s— ^ 2F2cos2a for the variable parameter a, we obtain the locus ACC\ whence it appears that the path described by each one of the bodies projected from with the velocity F touches the para- bola ACC\ If in the equation V^ — 2V'^gy—g'^x^ = we substitute for x and y their values u^t and v^t—'^ respectively, we obtain 76 EXAMPLE OF LEAST ACTION. [45. V'^—gv^t = 0, or ^ = — , giving the time from to the point of contact with the envelope, which is then positive if v^ be positive. The following conclusions may now be drawn : (i) If a point be taken outside the parabola ACC^ it cannot be reached by a body projected from with the given velocity F, because for such points being negative, the formulae above obtained give imaginary values for tj a, and A. (2) If a point be taken within the parabola ACC% it can be reached by a body projected from with the given velocity V in two different directions, J giving rise to two distinct courses in which the initial momenta, the times of flighty and the Action have different values, one of these courses (OCP) reach- ing P after touching the envelope, and the other (OPQ) either touching the envelope after passing Fig. 4. " through P, or not at all, according as the direction of projection in it is above or below the horizontal. If the direction of projection from be below the horizontal, the course touches the envelope at a point to the left of 0, that is at a point for which t is negative. (3) If P and Q (Fig. 4) be two points taken on one of the projectile paths, P before and Q beyond the point of contact (C) of that path with the envelope, then the initial momenta, time, and Action expressed in terms of the coordinates of P and Q respectively, will be each of them given by taking in the one 45-] EXAMPLE OP LEAST ACTION. 77 case the negative and in the other the positive value of the radical. This is made clear from the expression in (II) yx gx Whence it follows that tan a = — ) where x-^ is the abscissa of C\ But the abscissa at P is less than a?i, and that at Q is greater than x^^ and V is constant. It follows therefore that in the expression for tan a in terms of the coordinates of P or of Q, when the coordinates of P are substituted the numerator must be less than V^, and when the coordinates of Q are substituted the numerator must be greater than V^, i.e. the negative sign of the radical must be taken between and C\ and the positive sign must be taken beyond C\ It thus appears that neither the time^ the initial momenta, nor the Action is expressed hy one and the same function of the co- ordinates throughout the whole course. The function in each case expressing these quantities changes its form or type at the point of contact with the envelope. (4) Of the two courses from to P, that one which reaches P after touching the envelope has the greater Action. For, let t^j t^ be the times, A-^ , J2 ^^ Actions, in the two courses ; then from (III) Also from (^), Art. 41, above 2 = — Ah/. 9 78 KINETIC FOCI. [46. Therefore but since {t,^t, '!= -gVx^ + ' •gy-g V -ly^gy — g^x^ is positive, y'-gy > g VaF+^f. Therefore A^^A^ has the same sign as t^ — t^'j and since the course which reaches P after touching the envelope has the greater time, it has the greater Action. It follows that if A^^A^ be the two functions expressing the Action from to P, A^ — A^ is for real values of A-^ and A^ essentially negative if the course having Action A2 reach P after touching the envelope. (5) The curves of equal Action are as above proved normal to the courses, and therefore when they meet the enveloping parabola must be at right angles to it. It follows that the curve A — constant has two branches, forming a cusp when it meets the envelope, and one branch, the upper, intersects the courses orthogonally after they touch the envelope. 46.] JDejinition. — The point C to which the two courses from coincide is called a kinetic focus conjugate to 0. Evidently it is the point in which the courses touch the envelope. We may now shew that in the case of the projectile the Action from to P in the natural course is less than it would be in any infinitely near constrained course provided P lie between and (7, the kinetic focus conjugate to in the na- tural course OPC, where the Action changes type. Let P lie between and C, and let OM be a natural course, very near to OP, such that M may be reach- ed by a projectile starting from with the same energy as that in OP, and with in- itial momenta the same functions of the coordinates of Jf, as those in OP are of the coordinates of P, i. e. with the use in each case of the negative value of the radical spoken of in the Fig. 5. 46.] A KINETIC TRIANGLE. 79 discussion of the last article. At M suppose a fresh infinitely small impulse applied to the particle so as not to change its total energy and therefore not to change its velocity at that point but to cause it to reach P by another projectile trajectory MP, infinitely near OP. It is clear that any point M' infinitely near M in the course MP may be reached by a trajectory OM^ starting with the same total energy, therefore with the same velocity, as in OP and with the same type or sign of the radical. Let ji? and q be either generalised momentum and correspond- ing velocity at M in OM, and let jt?', / be corresponding quanti- ties at M in the course MM\ Let A be the Action in the course OM from to if, and A + hA the Action in the course OM' from to M\ Then, from the general proposition dA dq = P we have bA -. = '2pbq. Also hq = q ht if U be the time from 31 to 3r in the course MM\ Similarly 6'^,, the Action from if to M', = 2p'q'bt. Therefore Action 0M'< > Action 0M+ Action MM ' as 2^^'< > Sy^'. But since E and V are respectively the same at M in the courses OM and ifif', it follows that T must be the same in both courses, i.e. 2j?^ = 2^/. Therefore, by Proposition IX, Therefore Action OM' < Action OM + Action MM\ Similarly, if if" be a point in 31' P very near 31\ Action 031" < Action OM' + Action i/'i/ ". And so on, Action OP < Action OM + Action 31 P. It is clear that every constrained course from to P may be broken up into a number of natural courses, and that by the So A KINETIC TRIANGLE. [46. continued application of this proposition we shall always have — Action OP less than sum of the Actions in the broken course, or Action OF less than the Action in the constrained course. If now the point were taken on the course OFC, beyond C, as at Q (Fig. 6), we cannot make use of the previous reasoning, dA because the use of the equation -j— = p implies that the same sign of the radical is taken throughout in the expression for A. If then the Action in OM have the negative sign of the radical, i/ lying very near some point in the course be- tween and C, and if, as before^ we draw a series of natural courses from to points in MQ, the Action in all such courses must, in order that the proposition may be applied, have the same sign of the radical, that is the negative sign. But the Action from to Q in OCQ has, as we have seen, the positive sign. If then the constrained course MQ be so drawn as not to meet the envelope, the continued application of the proposition would result in proving that Action (OM+MQ) is greater, not than Action OCQ, but than Action OC^Q, the Action in the other, and as we have proved the shorter, course from to Q. But if the point M in Fig. 6 were so taken as that the Action in Olf in a course infinitely near to OCQ should have the positive sign of the radical, that is if M were taken beyond the kinetic focus Cj the proposition might be applied to shew that Action {0M+ MQ) is greater than Action OCQ. As we have already shewn that a natural course exists from to Q having less Action than OCQ, it is easily seen that some constrained course exists having less Action. For instance, let Q' be very near C and beyond it (Fig. 7), OC^Q' the course of less Action from to Q\ if a point in OCQ' very near Q\ Let the system receive at M any small impulse not altering its 47'] EXAMPLE OF KINETIC FOCI. 81 kinetic energy, so as to cause it to describe a new trajectory in- finitely near OCQ^Q and intersecting OCQ'Q in Q, Then by the above method it may be shewn that Action {3IQ'-\- Q'Q) > Action MQ, .-. Action {OQ' + Q'Q) > Action (Oif + MQ), .-. a fortiori Action OCQ > Action {031 -^MQ); and 0M-\- MQ is a constrained course infinitely near OCQ. It appears therefore that in the case of the projectile the Action from to any point in the course is a true minimum, so long as it is represented by tke same function of the initial and final coordinates, and ceases to be a minimum when the function changes. 47.] An additional illustration of this subject may be derived from the second of the two examples investigated above, namely, that of the ellipse under the action of a force varying as the distance. In that case we have seen that 8= ^{(a7'^ + 2/' + < + 2/o')cotyM«-2(a;a?o + 2/2/o)cosec>v/iIO» W and that , o . i^ ^' + y' + ^o' + 2/0^-2 (^^0 + 2/2/0) cos y^^ . /TJX where in (II) we must substitute for t the value given by the equation cos V'm^ = 2^(^^0 + 2^2/0) ± a/ I - ^ (^ + 2/^ + < + 2/oO+ 5^ (^>o + 2/2/o)^ .. (HI) E being the total energy. The value of A thus obtained will be the total Action from Q 82 EXAMPLE OF KINETIC FOCI. [47. any given initial to any final configuration in terms of the coordinates of the two configurations. The differential co- efficients of A with regard to x^ and t/^ respectively, after the required substitution for t, give us the requisite initial momenta to enable the particle to pass from the initial to the final con- figurations in terms of the coordinates of those configurations. It appears from (III) that there are two distinct values of cosy/ju^ in terms of the coordinates of the two configurations, and therefore two distinct elliptic orbits, by either of which the particle may move from 0, the point of projection, to P with given initial kinetic energy. Again, in either orbit the motion may be either direct or retrograde. In each ellipse the value of cos ^/ixt is the same for the direct as for the retrograde motion, but \/yLt is represented in the one case (which we may call the direct motion) by 0^ and in the other by 277— ^, 6 being a positive angle less than 77, and having a single value for every point in the orbit. Regarding only the two direct motions from to P, we shall obtain two distinct values of each initial momentum, and of the Action^ i.e, two courses expressible by two distinct functions f-^ and f^ of the initial and final coordinates ^. Similarly, if we regard the two retrograde motions, we shall obtain two other distinct courses from to Pj expressible by the functions ^3 andj'^ of the initial and final coordinates. The two ellipses will coincide when the values of x, y cause the radical in (III) to vanish, and in that case the two direct courses coincide, and likewise the two retrograde courses. This locus oi x^y is clearly an ellipse, and if the starting-point be taken at a distance c, from the origin on the axis of x so that a :=.c, a =^^ and if the initial velocity be V so that 2E= Y''-\-\ic\ the locus easily reduces to * We here neglect all the other values arising from the expression t = cos—* m when m is known, because these only correspond to the return of the particle to the point +x,±_y after successive revolutions. 47-] EXAMPLE OF KINETIC FOCI. 83 If we express the integrals of our equations of motion in the form X ^i a cos ( \//ix t + a), y = 6'sin(>/)Lii{), we find that the particle describes an ellipse whose equation is a^ 2s>maxy y^ c? cos^ a V a cos^ a 6'^ cos^ a ~ ' where F^ = ju (a^ sin^ a + V) ; and the coordinates ^q? ^o ^^ ^^® point of projection are ^cosa and ; putting c for cCq the equations become x^ 2ianaxy y^ _ r = iu(cHan2a + 6'2) (B) If we investigate the envelope of (A), with V and a variable parameters subject to the condition (B), we obtain, as we should expect to do, the aforesaid equation x^ y^ 1 If 0?, y be the point of contact of (A) with this envelope, i. e. the kinetic focus conjugate to the point of projection c^ in (A), we get y y , ^' — cota- X c V^ + fJ^c^ If fl^ be or 90°, i.e. if the point of projection be at the ex- tremity of one of the principal diameters of the ellipse described, we get either ^ = or 5^ = at the kinetic focus, shewing that this focus is situated at the extremity of the other principal diameter, which therefore, as will be shewn later, is the last point to which the Action is a minimum. It is worth remarking that both in this problem and that of the projectile, the direction of motion at the point of contact with the envelope is at right angles to the direction of motion at the point of projection. We may now draw a very similar series of inferences with reference to this problem to those drawn in the case of the projectile, namely : G 2, 84 EXAMPLE OF KINETIC FOCI. [47. If a particle be projected from the given position c, with given velocity F, and it be required to pass through another assigned position, then if this assigned position lie within the ellipse 2/' 1 F^T^»+ F"^ = ^' ^^^ there are four distinct directions in which the particle may be projected so as to pass through the second position, that is to say, two for direct and two for retrograde motion. If the second position lie without the ellipse (C) it will be impossible to project the particle so as to pass through this second position. If the second position be upon the ellipse (C), there are for the direct motion two coincident directions of projection, and similarly two for the retrograde motion, and the ellipse described by the particle touches (C) at the second position. The ellipse described by the particle always touches (C) either before or after passing through the second position, and the type of the motion, that is to say, the functions of the initial and final coordinates giving the requisite initial and final momenta, and the Action, changes at the point of contact of each trajectory with the envelope (C). It may be proved, as in case of the projectile, that of two courses to any point P that one which reaches P before touching the envelope has the less Action. But as this is proved subsequently by a ge^neral method applicable to all cases, it is unnecessary to verify it in the special case of the ellipse. There will be another point of contact with the ellipse C, and therefore another kinetic focus and change of type in the second half of the orbit. Again, when the particle arrives at the extremity of the diameter through the point of projection, J \xt = tt ; that is, Q= 2-17— 6 = tt; and the Action there, as is easily seen and will be proved in the sequel, changes type by the adoption of 277—6 instead of as the value of ViJtt in its expression. And on again passing through it changes type by the adoption of 2tt-\-6 for 277 — ^, so that there are in fact four changes of 48.] LEAST ACTION. 85 type in each complete revolution, namely, two at the points of contact with the enveloping ellipse^ and two at the ex- tremities of the diameter through 0. 48.] Analogous propositions to those which we have thus established for special cases can be proved for the general case of any conservative system, having any number of degrees of freedom, acted on by forces continuous functions of the co- ordinates, and moving from a given initial configuration with the sum of its potential and kinetic energies equal to E. For let us consider any conservative system with any number, %, of generalised coordinates qx^%-> •••$'nj indicated generally by ^y and let this system be acted on by any given forces. Suppose the system to be initially in any given configuration in which the coordinates are indicated generally by ^g, and to be started from that configuration with total energy E. Let the initial configuration q^ be represented by the point 0, and the final configuration q by the point P, and let the series of intermediate configurations through which the system passes be represented by the points in the curve OCT ; then OCF represents a course or motion of the system from q^ to q"^. If we attempt, as in the case of the single particle hitherto treated,, to express each initial momentum at in terms of the initial and final coordinates and energy, q^^ q and E, we shall generally find, as we found in case of the particle, that each of these initial momenta will be expressed by a function of the above-mentioned variables having a plurality of forms or values, such as ^/^,(?a.^,E), x/.,fe,2,E)&c., corresponding generally to as many distinct courses or routes by which the system can move from to P. The time from to P, as also the Action, will be expressed by functions having a similar plurality of form. * It will be understood of course that the curve OOP does not represent the motion of the system from the initial to the final configuration in the same way as in the case of the single particle, because each configuration involves many coordinates which cannot be thus graphically denoted. The length of the course must be measured by the time from one configuration to another, as before explained, and is only inadequately represented by the curve joining the points indicating such configurations. 86 LEAST ACTION. [49. It may be, as in the ease of the ellipse before treated, that these functions, or a class of them, although comprehended imder one general form, yet contain in their expression a func- tion having many values, as for instance cos~'^m, where m is a single- valued function of the coordinates, and differ from each other only by attributing diflPerent numerical values to that function. We treat these functions as having different types according to the different values given to the function in question. See Art. 56, post. In the case of the Action, with which we are now chiefly concerned, these forms will be henceforth denoted by /iteo.^, E), f,{q„q,E), &c. or shortly ^^^,^2 ^c* 5 ^^^ ^^ ^^^ initial coordinates q^, and also E, are supposed invariable, these symbols may be regarded for our present purpose as functions of the final coordinates q only. 49.] It may be that for certain values of the final coordinates, that is for a certain final configuration S, two functions ex- pressing the initial momenta, such as \//-i, x}/^, become equal in value for each one of the momenta. In that case two courses from to S become coincident. The configuration S is then defined to be a kinetic focus conjugate to the configuration 0. Inasmuch as there are n initial momenta, this equality gives at first sight n equations for determining the n coordinates of 8 in order to satisfy the condition. But it must be remembered that, E being given, any one of the initial momenta may be determined as a function of the remaining n—\ and E, so that in fact, of the n equations expressing the equality of the initial momenta of the given types ^\r^ and i/^g, only n—\ are independent. They are not then generally sufficient to deter- mine a single position of /S, but determine a series of such positions constituting a quasi locus or envelope in many respects analogous to the envelope in the cases of the projectile and ellipse. And among other things, this quasi locus or envelope has the property that configurations properly situated with regard to it cannot be reached by the system starting from with momenta of the y]r^ or -^^ type. Whenever two types^ as \/r^ and -^^^ become equal in value for 50.] LEAST ACTION. 87 every one of the initial momenta, the corresponding courses from to 8 become, as above mentioned, coincident, and there- fore of course the two corresponding functions expressing the Action become equal in value. But the converse is not true ; for two types of the Action, as/j and /g, may for certain final coordinates be equal in value, while the corresponding functions expressing the momenta remain unequal. In that case two non-coincident courses have equal Action from to the final configuration P. 50.] It appears then that the most general case presents the following analogies with the case of a single particle, viz. (i) If the final configuration P be arbitrarily chosen, there are generally a certain number (say r) of courses by which the system may move from to P, these courses being determined by the types of the functions of the coordinates of P selected for expressing the momenta at 0. (2) For certain final configurations any two of these courses may become coincident ; for others they may become impossible. (3) It was proved in the case of the projectile that the function of the final coordinates expressing the Action from the point of projection changes type at the kinetic focus, or point of contact with the envelope. And in like manner, as we proceed to shew, the function expressing the Action from the initial con- figuration in any conservative system changes type as the system passes through a kinetic focus conjugate to the initial con- figuration. (4) It appeared in the case of the ellipse that the Action changes type at the completion of the half period. In like manner we shall shew that if any conservative system, being set in motion, returns by a natural course to the configuration whence it started, making a complete circuit, the Action changes type at the completion of the half circuit. (5) It was further proved in the case of the projectile that the Action in the natural course from the point of projection to any point P reached before the change of type is necessarily less than the Action in any infinitely near constrained course from to P, and is therefore a true minimum, but if P be a point in 88 LEAST ACTION. [5 1. the course reached after the change of type, then the Action is not necessarily less in the natural than in the constrained course. Analogous propositions will be proved true for any conservative sj^stem. 51.] Let the system move from in a course OC-^^8 ... (Fig. 8). Let P be any configuration through which the system passes at the time t. Let the Action in that course from to P be represented by the function,, j^, of the coordinates of P, so long as P lies between and a certain configuration S in that course : and at S let f^ =^2^5 f^ ^eing another of the functions ex- pressing the Action from 0. Then if there exist real courses from to P having Action of the type f^ for all positions of P between and S, or C-^ and S, it can be shewn that f^ > f^ for all such positions of P. For let OC^ denote a distinct course from to P in which the Action has the type f^ . Fig. 8. Then/i and/2 are both functions of the coordinates of the final configuration P, and as such change with the time t as the system moves on in its course 06\P Therefore if ^, p denote the velocities and momenta at P in the course OC^P..., and q,p' those in the course OC^P, we have * We use the expression /j at S as an abbreviation for /i ivhen the final coordinates are those of S. 53-] LEAST ACTION. 89 Now since E^ the total energy, is the same for the two courses OCJP and OC^P, the kinetic energy at P is the same in both courses ; that is, 2qp = ^q^p\ and therefore ^qp — ^qp\ or 7 -j-Afi—f^y is necessarily positive (Art. 9, Prop. VIII). There- fore since f<^ ~f\ at S^ f^ >f^ if P be reached before S. It thus appears that/i, the Action in the course, always increases faster than /g^ a,s the system moves on in the course OC^..., This is a particular case of the theorem of Art. 10. 52.] Next, let P' (Fig. 8) be a configuration in the course OCiS... beyond S. In that case, remembering the result ob- tained for the projectile, we do not know whether the Action in OC^SP^ has the type y^ ^^ f^i inasmuch as it may change type at 8, But whichever type it has, let OQ^P' denote a course from to P' in which the Action has the other of the two types in question. Then the process of the last Article shews that Action OC^ /SP' > Action OC^P", so that Action OC^SF has the type f^ ox f^^ whichever is the greater. We see then that when the system moving in its course OC^,., passes through S, where /^ =^2^ ^^^ ^^ ^^^ things must happen, viz. either (i) fi^f^ changes sign, or (2) the Action in the course changes type. 53.] Three distinct cases have now to be examined. Firstly, S may be the first kinetic focus conjugate to in the course OC^..., and therefore such a configuration that not only two types of Action, f^ andj^g? become equal, but also that two types of functions expressing the initial momenta become equal when the final coordinates are those of S. In this case there are two coincident courses from to S. Or secondly, S may be a configuration at which only two types of the Action become equal, and therefore may be represented by the point of intersection of two non- coincident courses having equal Action (Fig. 9). Or thirdly, the momenta at 8 in 00^8 may be respectively equal and opposite to those in OC^ 8. In that case the system, in whichever of the two courses it be started from 0, returns again to 0, so that the two courses are coincident but are 90 LEAST ACTION. [54. described in opposite directions. This case evidently includes all periodic motions. Fig. 9 54.] Tq 23rove that if 8 he a kinetic focus^ the Action must change t^pe at 8. Let /S be a kinetic focus, Q any configuration in OC^S infinitely near S and beyond it. Then the Action in OC^SQ has one of the two types/i,y2j which become equal at S. Let OC^QS' be a course in which the Action has the other of those two types. Fig. 10.* Then since OC.^ QS' denotes a motion infinitely little varied from OC^^S^ there must by the continuity of the motion generally be a * The dotted line OS in the figure indicates the second course from Ui S. It would in fact completely coincide with OCiQ'S. 55-] LEAST ACTION. 91 kinetic focus corresponding to S somewhere in OCc^QS'. But there can be no such between and Q : for if there were we could prove by the proposition of Art. 52 that Action OC2Q is greater than Action OC^SQ, whereas we have above proved it to be less. The kinetic focus in OC^ QS' must lie at S^ beyond Q. Let Q' be a configuration in OC-^^8 very near S and between and 8. Then the Action from to Q in OC^ Q must by the continuity of the motion have the same type as the Action from to Q' in OC^Q\ But Action OC^Q has different type from OC^Q, for if it had the same the two courses would coincide, which is not the case. Therefore Action OC^Q^ has different type from OCiQ. And as this is true however near Q and Q' may be to S, provided they do not absolutely coincide with it, and are on opposite sides of it, it follows that the Action in OC^S must change type at S, 55.] If, on the other hand, S be not a kinetic focus, the two courses OC^S and OC^S (Fig. 9) are not coincident, and the momenta at S in OCj^S are not all equal to those in OC^S. Therefore by Art. 9, Proposition VIII, not only is ^qp — ^qj/ positive, but also it is not zero. That is, -j- {fi—f^j is not zero, and therefore j^ —^2 generally changes sign at S^ and the Action does not change type. An important exception occurs in the third case above re- ferred to, when the momenta at S in 00^,8 are equal and opposite to those in 00^8, and consequently the system, by whichever of the two courses it be started, returns to com- pleting the circuit. This case includes all periodic motions. It may be considered as a case of two coincident courses described in opposite directions. We shall find that the Action in such cases changes type at 8. For if j^ be the type which it has at starting, /^ is zero at the beginning of the circuit, and, being a function of the final coordinates, must also be zero at the end as the system returns to 0. Therefore^ cannot increase with the time throughout the circuit. But so long as /[ continues to be the Action in the course, ^ ■=.%T^ and 92 LEAST ACTION. • [56. therefore f^ must go on increasing with the time at a finite rate. We see then that the Action must change type some- where in the circuit, and that can only be when /i —f^^ that is, at S. It must therefore change type at S^ and —^ must at (it that instant change sign discontinuously. An example of this occurs in the case of the elliptic orbit above discussed, where the Action to /S', the extremity of the diameter through the point of projection^ has the same value for the retrograde as for the direct motion. And as we said, it there changes type, adopting for the second half of the orbit i-n—Q instead of Q in its expression, that is the greater instead of the less value of \/[Lt derived from III of Art. 47. So generally, if the Action in different courses from io F depends upon the different values of 2/77+^, where 6 is a single- valued function of the coordinates of Pj and i an integer, the Action changes type whenever 6 is zero or rr, although the configuration where that occurs may not be a kinetic focus. We have thus established the propositions contained in (3) and (4) of Art. 50. It remains to establish that contained in (5). 56.] If a configuration P he taken on any course 0C^8 starting from 0, between the configuration and S, the configuration at which the Action first changes type^ the Action from to P in the course OC^P will he less than that in any i7ifinitely near con» strained course ; hut if the configuration P he taken heyond S, the Action in the statural course OC^SP will not necessarily he less thayi in the infinitely near constrained course. For if M be any configuration not in the course but infinitely near some configuration in the course between and P, it will be always possible by the continuity of the motion for the system to move from to ilf by a natural course of the original type, that is, in which the Action is f^. At M let the system receive small impulses altering the direction of motion but not the kinetic energy, so as to make it pass to M.\ and so on by the constrained course MM'P from M to P, being always infinitely near OC^P. 56.] • LEAST ACTION. 93 Then also it will be always possible for the system to move by a natural course OM' having Action of the typeyi, from to 3I\ any configuration between M and P in the course MP, Let 3f be infinitely near M. Fig. II. Let the coordinates o^ Mhe q^... q^y those of If' ^i + 8^i, &c. Let the action in Oif be /, that in OM'f+hf. Let ^, q denote the momenta and velocities in OM, p\ q those in MF. Then, hf=^^-Ih Action 01/'. Similarly, if M" be any other configuration in l/'P infinitely near if'. Action Oil[f' + Action if 'J[f" > Action 031" ; and by the continued application of this method we prove that Action 0M-\- Action 3IP > Action OP ; and since M may be as near as we please, the proof applies to 94 LEAST ACTION. [57. any possible constrained course from to P, infinitely near OC^P, and having the total energy E. The above process would fail if P were on the other side of S, and therefore the Action in OSP were of a different type from/i. For in that case it would, as in the case of the projectile, gene- rally result in proving that Action {031+ MP) is greater,, not than Action OC^P, but than Action OC^P,^ a course from to P having the same type of Action as OM. And this holds if the change of type consists in giving different values to a multiple function such as cos~^^ (see Art. 48). We should in that case prove that Action [OM+MP) > Action OP only where we use both in 031 and OP the value of cos~^ m which is less than it, or in both the value between it and 2 it, and so on. Suppose a material system can move from to P, from P to Q, and from Q to 0, the total energy being the same throughout but the momenta at 0, P, and Q different in two adjacent courses. If we define OPQ so constituted to be a kinetic triangle, the Actions in OP, PQ, and QO its sides ; the process of this Article shews that two sides of such a triangle are necessarily greater than the third side, provided the Action in either of the two sides be of the same type with that in the third side ; otherwise not necessarily. 57.] We may therefore draw the following conclusions applic- able to any conservative system whatever : — (i) So long as the Action from in any course retains the same type, it is less than the Action in any infinitely little varied course from to the same final configuration, and is therefore a true minimum ; (2) But generally ceases to he the least possible when a con- figuration is passed to which any other course exists having equal Action. (3) After the change of type the Action ceases to he a minimum, * OC'P is not shewn in the figure. 6o.] LEAST ACTION. 95 58.] The following" is the analogous theorem in Geometry : — If be a point on a surface, and geodetic lines be drawn from it; and if S be the point of ultimate intersection of two such geodetics, OC^S, OC^S, when they very nearly coincide, then if P be any point in OC-^^S between and S, OC^P is shorter than any line from to P that can be drawn upon the surface infinitely near it, but not necessarily shorter than any line whatever drawn on the surface from to P. If P lie on OC-^S beyond S^ a line can be drawn upon the surface from to P infinitely near to and shorter than OC^SP, 59.] Now let the system having passed 8^^ the first kinetic focus, and there as we have seen acquired /g for its type of Action, arrive at a second kinetic focus 8^^ where f^ —f^- It can then be shewn, exactly as in Art. 54, that the Action again changes type, and becomes/*3 for configurations beyond 8^ . And in like manner the system may successively assume all the types /i . . .f^ from the least to the greatest. 60.] As an example of a kinetic focus occurring in a system of many degrees of freedom we may take the case of a system of projectiles. Let, for instance, A material particles of masses respectively m^..,mx be simultaneously projected in the same vertical plane from a point 0, the sum of their kinetic energies at being equal to a given constant E. Let it be required to find the initial velocities of the several particles in order that the system, so started from 0, may pass through a given configuration ^j, y^, ... a?^, ^aj that is, that the particle m-^ may be at x-^^ y^, and m^ at a?2, y^^y and so on all at the same instant. Let be the origin of coordinates, the axis of X being horizontal in the plane of projection. Let % . . . ^A be the horizontal, v-^...Vx the vertical velocities of the several particles at 0. Let t be the time measured from the instant of projection. Then 2E = 2m(w2 + v2). Also ^1 = -r ' Wj = -^ , &c. ; t t and therefore 2 m w'' = 2 m — • t 96 LEAST ACTIOX. [6o. Also y^=^v,t-\ge, or v,=i^ + ^gt) similarly v^ = ~ +\gt, &c. = &c. Therefore ^mv^ ='2m~ ■{■^m- — +g2my. And 2E = ^rn-^ +Sm ^ +^2m2/. Whence we obtain — ^ = 2E-^2m2/ It appears then that there are two distinct values, two equal values, or no possible value, of f^, according as the quantity under the radical sign in the above expression for is positive, zero, or negative. Again, for any particle, as m-^^y which for either value of f^ gives a single value of u^-\-v^. Similarly which also gives a single value of— for each value of P'. There- fore for each value of P' there is a single possible course for each particle. It follows that corresponding to the two values of P there are, if the quantity under the radical sign be positive, two, and only two, distinct courses by which the system can pass from to the given configuration x^^y^ ■ • • ^a? ^a- When the quantity under the radical sign becomes zero, the two courses become coincident, and the configuration x^^y^ ... is then a kinetic focus conjugate to 0, Again, by substituting nt for x and vt—\gP for ^ in the quantity under the radical, it will be reduced to {2^—gt Imvf^ 6 1.] LEAST ACTION. 97 Therefore the time at which the kinetic focus is reached is found 2 F from 2E—gtl,mv = or ^ = -— — . This is necessarily posi- g^mv '' ^ tive if ^mv be positive, that is if the direction of motion of the centre of gravity of the system at be above the horizontal line. In that case the system necessarily passes through one and only one kinetic focus after projection from 0. The theorems of Arts. 51 and 54 may be verified in this case as in that of the single projectile. 61.] It is evident that ar kinetic focus may be regarded as the ultimate intersection of two neighbouring courses from the same initial configuration 0, both having the same value of E, the sum of the potential and kinetic energies, when the initial momenta in the one motion differ infinitely little from those in the other. If the system while in the initial configuration receive im- pulses changing its momenta from jo^ . . . j9„ to jt?i + 'bp-^ . . • it?„ + c)^„, such impulses are said to constitute a disturbance of the motion ; and if the variations of the momenta are such as not to alter the kinetic energy of the system in the given configuration, the disturbance is called a conservative disturbance. We may obtain a general equation showing the condition that a kinetic focus may exist in a given course from 0, and at the same time determining its position if it exist, and the nature of the disturbance in order that the disturbed course may intersect the undisturbed one, i. e. have a configuration in common with it. Thus — Let ji?! . . . Pn be the initial momenta at 0. Then E being given, any one of the j&'s, e.g. jo„, may be expressed in terms of the others and E, so that only^?! ...Pn-\ are independent. Let / be the Action in the original system from to a configuration P whose coordinates are §'i . . . ^„ ; and let the Action in the varied motion from to P', whose coordinates are ^i + c>^i, &c., have the same value/. Then, p denoting the momenta in the original course at P, 98 LEAST ACTION. [6 1. This constitutes one relation between the variations <)^i...<)^„, from which any one, e.g. ^^„, may be found in terms of the others, so that only bq^^ ... hq^-i are independent. Now qi ... qn ^"^^ functions of pi...Pn ^^^ A ^^^ ^^j ^^ we have seen, of^, ...Pn-if E, and/. If therefore, / being constant, -^ stand for (-^) + -j-^ -7— dp^ ^dp/ dpn dj\ ("^ being" the partial differential coefficient of ^„ with regard \dp^ to^i when7?„ is expressed as a function of ^^ •.•i^«-i ^^^ ^) ^"^ the. other coefficients -$- in like manner, we have, in order to determine ^q^... ^$'„_i, the system &c. = &c. ; in which the coefficients -^ are functions of /?i.../?„_i, E, and/. dp In order that P may be a kinetic focus conjugate to 0, every "bq must vanish otherwise than by the vanishing of ^^^ ... ^Pn-\^ if the proper value of /be substituted in the coefficients — , and proper values given to the ratios of the <)/>^s. But this cannot happen for any set of ratios unless the determinant of the system be zero, that is, 2±(^.../^) = (B) This then is an equation from which / the action from to P, and thence also the time t of reaching the kinetic focus, may be determined. If it have one or more real and positive roots dif- fering from zero, each of them corresponds to the time at which the system, started from with the momenta />j ... Pn-u reaches a kinetic focus conjugate to 0, and therefore determines the position of that focus. 6 1.] LEAST ACTION. 99 In the case of the projectile or any other motion in a plane curve, the curve of equal Action is normal to the course. If the undisturbed course intersect it in P, and the disturbed one in F, then let PF=z Tig, or if Tip be the variation of one of the two initial momenta, PP" = -^'bp. dp ^ In order that P may be a kinetic focus we must have -^ = 0, that being the form which the equation (B) assumes in this case. Further, in case there be more than two degrees of freedom, if the equation (B) be satisfied, it harmonises the equations (A), and they then suffice to determine the ratios which <^pi . . . ^Pn-i must bear to each other, that is the particular kind of disturbance, in order that the disturbed and undisturbed courses may have a configuration in common. If for different roots of the equation (B) these ratios have different values, they correspond to distinct disturbed courses from 0, each of which intersects the original course. If two or more roots of (B) correspond to one and the same set of ratios of 'bpi ... '^Pn-u then the same disturbed course intersects the un- disturbed course more than once. The second and subsequent intersections may with propriety be called secondary kinetic foci. In the elliptic orbit before considered, the disturbed course inter- sects the undisturbed one four times in each complete revolution. In the case of the projectile, on the other hand, the two courses, having once intersected each other in the kinetic focus, will never after again intersect. If in equation (B) we were to substitute for Pi ... Pn-i and f their values as functions of the initial and final coordinates, (B) would then be an equation between the final coordinates involving the initial coordinates as constants ; the equation namely to the locus of kinetic foci, or envelope of the system. H % CHAPTEE V. APPENDIX. Article 62.] We referred in Article 1 to an expression for jp^, the generalised component of momentum corresponding to the coordinate q^ in the language of quaternions, viz. the scalar function v./ J'? ^9 at dq^ p being the vector from the origin to an element of the system of mass m, expressed as a function of the n scalar variables qi ...q„, and 2' denoting summation for all such elements. In like manner dt^ dqr denotes the generalised component of force, F^, corresponding to the coordinate q^. If we denote by P^ and G^ respectively the corresponding vector functions ^, dp dp , v/ TT ^V ^P dt dqr dr dq^ it will be found that P and G^ possess analytical properties similar in many respects to those already investigated for p and jP. For we have dp dp . dp . dp . substituting which in the expressions for jp and P respectively, we obtain VECTOR COMPONENTS OF MOMENTUM. 101 ^--'»'^^S^>- -^'-^w-^ in which every -^ is a vector. Evidently in the expression for Pr the term involving ^^ disappears, because If we write these equations in the form the coefficients a are all scalars, and the coefficients b are the corresponding vectors. Further, , dp dp dp dp dqr dq, dq, dq^ as already proved, but ^ dp dp dp dp _ dqr dq, dq, dq^ If ^'i... qn ^e any other set of component velocities which the system might have in the same configuration, we shall obtain from the above as above proved, but 2 denoting summation from 1 to n. Also 2P^'= 6,2 {^2 ii- S'l f 2} + &13 {^3 2'i- ix i'3} + - and by making q = ^, we obtain 2P^ = 0. Again, Lagrange's equations may be written -d-t-i^^d-q^^^ = ^^^dq-^'^^Wrq' to which corresponds dP ^^.dP ^, „d^p dp ,^.dP ^ 17 = i^*^+^'»''^4=*^*^-'^'- 102 THE VIRIAL. [63. 63.] The scalar function — IfmS^p is equal to -,, , doc dy dz. ■, ^ ^/ / o 9 o\ ^^mV-^p is twice the vector area described by the system per unit of time about an axis through the origin. In Cartesian coordinates ^'m(x~- —y-ri) is twice the area ^ at ^ dt^ described per unit of time about the axis of z. If a line be mea- sured off along that axis representing in magnitude i dy dx\ \ dt ^ dtS and if the corresponding lines be taken for the other two axes, and the resultant of these three lines be formed, that resultant is represented in magnitude and direction by ^^mV — p. dt If this be denoted by A, and if the corresponding scalar -,, ^dp ., . . , ^, ( dx dy dz) -S m^-p, or Its equivalent 2 m|a.- +3,- +^^ |, be denoted by S, we see that P stands in the same relation to A in which J!? stands to S, so that dS ^ ^ dA p = -— and P = —— , dq dq the actual velocities w, y, z being in either case regarded as constant. 64.] We conclude with Clausius' theorem on the equality of the mean kinetic energy and the mean virial, as expressed in generalised coordinates. In the expressions obtained in the last Article, if for the linear velocities * dp dx dy dz -7-3 or ^r-3 -^J -rr-i dt dt dt dt we substitute the effective accelerations d'^p d^x d'^y d^z df" ' ^'" dt"^ ' dt" ' de ' the scalar S becomes YmS^p or '2fm{Xx-\-Yy-^Zz], 65.] THE VIRIAL. 103 the half of which is called the virial of the system, which we will denote by V. The vector A becomes the moment of the re- sultant couple. Now let us suppose that the nature of our material system is such that the mean value of 2'w^-^p is constant, if the time for which the mean is taken be sufficiently great. That is evidently the case for every strictly periodic motion if the means be taken for the periodic time; and it may be the case for motions which are not strictly periodic if a sufficiently long* time be in question. Any such motion may be defined to be stationary. As the expression ^'m — p has both a scalar and a vector part, both must be separately constant on average of the time in question, or, which is the same thing, both d -,. ^dp -zr^ nijS-zrP = on average. dt dt^ ^ . , d ^, dp ^ ^ (^) and -7--^ mV —r p =z on averaj^e. dt dt ^ *= The first of these equations gives or the mean kinetic energy added to the mean virial is zero. The second of equations (C) expresses the principle of con- servation of areas. 65.] If now p be such a function of ^j ... ^„ as that then ^^^mS — p = ^pq, dt and ^'mV^p = ^Pq, at so that in stationary motion both ^pq and '2,Pq are constant on average. And in this case 72 104 THE VIRIAL. [66. and is therefore identical with the virial as hitherto defined. 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The Delegates of the Press invite suggestions and advice from all persons interested in education; and will be thankful for hints, dfc. addressed to the SECRETARY TO THE DELEGATES, Clarendon Press, Oxford. Bifeles iPrinteb at ti)e (ttlatenlron |3rcss. THE OXFORD BIBLE FOR TEACHERS, THREE NEW EDITIONS, ON INDIA PAPER, EXTREMELY THIN AND LIGHT. IS. and professions of obedience. luicken thou me according to thy )rd. 6 I have declared my ways, and thou ardest me : a^ teach me thy statutes. 7 Make me to understand the way thy precepts: so J/ shall I talk of y wondrous works. 8 » My soul 2 melteth for heaviness : rengthen thou me according unto thy )rd, 9 Remove from me the way of lying : d grant me thy law graciously. I have chosen the way of truth: « ver. 12, Ps.. 2". 4. i 27. 11. *86. ]1. !/ Ps. 145. 5, C. r Ps. 107. 26. 2Heb. druppeth. No. 2a. Minion 8vo. THIN. (SUPEBINTENDENT'S EDITION.) Size, 73 inches long, 5i indies broad, and 1 inch tfiick. Weight 22 ounce.^. Paste grain morocco, limp . . . 13 6 Persian morocco, limp , . . . 16 6 Turkey morocco, limp . . . . 18 Turlvey morocco, flap edges . . . 110 Levant morocco, lined calf, flap edges . 14 Ditto, very flexible, silk sewed, red) under gold in the round— the mo^st > durable binding extant ) With Apocrypha, extra .... With Prayer-Book, extra 1 11 6 PSALMS. and professions of obedience. 1 LoED ; endur- aied in the law eep his mwith " quicken thou me according to thy word. 26 I have declared my ways, and thou heardest me: * teach me thy statutes. 27 Make me to understand the way of thy precepts: so ** shall I talk of thy wondrous works. 28 ^My soul 2 melteth for heavi- ness: strengthen thou me according unto thy word. 29 Remove from me the way of lying: and grant me thy law gra- ciously. 30 I have chosen the way of « Ter. 12. 1 Ps. 25. 4. A 27. U. & bC.. 11. I » Pa. 145. 5,6. • Pb. 1U7. 26. I 2 Ileb. I PSALMS. and professions of obedience. to the LoBD ; lercy endurelh hat keep his eek him with niqulty: they " quicken thou me according to thy word. 26 I have declared my ways, and thou iieardest me : * teach me thy statutes. 27 Make me to understand the way of thy precepts : so V shall I talk of thy wondrous works. 28 «My soul 2 melteth for heavi- ness: strengthen thou me according unto thy word. 29 Remove from me the way of lying: and grant me thy law gra- ciouslv. 30 1 have chosen the way of truth : Pi. U3. n. X T• durable binding extant ) With Apocrypha, extra .... With Prayer-Book, extra 7 8 10 12 14 6 18 2 3 2 3 Specimen leaves icill be sent on application. THE OXFORD BIBLE FOR TEACHERS. of obedience. D thy t. ver. 40. Pit. 143. IL thou ites. way Ik of Ps. 2-.. 4. 4 '27. n. *«6.11. y Ps. 145. 5,6. uess: f Ps. 107. Six Editions pointed on hest lUig-made pointing jQaper. No. 1. Minion Small 4to. (9| X 7 X 1| inches.) — Width of Margin. — A Superb Edition, with Wide Margins for Manu- script Notes. Cloth boards, red edges . . . . 12 Persian morocco, limp, red under gold edges 18 Best Turkey morocco, limp . . . 14 Best Turkey morocco, circuit edges . 1 10 Levant Morocco, lined calf, with flap edges 1 16 IS. and professions of obedience. luicken thou me according to thy )i-d. 6 I have declared my ways, and thou aidest me : « teach me thy statutes. 7 Make me to understand the way thy precepts: so 1/ shall 1 talk of y wondrous works. « ver. 12. Vf.. in. 4. * 27. 11. 4 86. 11. y Ps. 145. 5, 6. No. 2. Minion Crown 8vo. (7| X 5 j X 1| inches.) Cloth boards, red edges .... 8 French morocco, gilt edges . , . 10 Paste Grain morocco, limp . . . 10 French morocco, circuit edges . . 12 Best Turkey morocco, limp ... 15 Best Turkey morocco, circuit edges . 19 Levant morocco, calf lined, with flap edges 1 1 lALMS. and professions of chedience. in DALETH. 25 * My Soul cleaveth unto the dust : "quicken thou me accoruiug to thy word. 26 I have declared my ways, and thou heardest me: * teach me thy statutes. 27 Make me to understand the way J ] of thy precepts : so >' shall I talk of i thy wondrous works. Pb. ;«. 4. 4 i.7. 1 1. 4ttU. II. »P».Ha.5,6. No. 3. Nonpareil 8vo. (7x4|Xl^ inches.) Cloth boards, red edges . French morocco, gilt edges Paste Grain morocco, limp French morocco, circuit edges Best Turkey morocco, limp Best Turkey morocco, circuit edges Levant morocco, calf lined, with flap edges •* ALMS. and pi-ofessions of obedience. ) ; I " quicken thou me according to thy I- I word. I 26 I have declared my ways, and j thou heardest me : * teach me thy ^ statutes. ' I 27 Make me to understand the way ' of thy precepts : so ^ shall I talk of n I thy wondrous works. • ver. 12. Pa. 2.=). 4. t 27. 11. A 86. 11. »Pi.M5.5.6, rSALMS. and professions of obedience. Lord ; eudur- " quicken thou me according to thy word. 26 I have declared my ways, and thou heardest me: =^ teach me thy statutes. 27 Make me to understand the way of thy precepts: so S'sliall I talk of thy wondrous works. . " TOr. V2. Ps. 25. 4. 4 27.11. 4 SC. U. I ► Ps. 145. 5,6. PSALMS. and professions of obedience. to the Lord ; lercy ettdureth mdry prayen, qf (wedienct. " quicken thou me according to thy word. 26 I have declared my ways, and thou heardest me: * teach me thy statutes. 27 Make me to understand tlie way of thy precepts: so V shall I talk of thy wondrous works. No. 4. Red Line Edition. Nonpareil 8vo. (7 x 4| x 1| inches.) Persian morocco, red under gold edges 12 Best Turkey morocco . . . . 15 Best Turkey morocco, circuit edges . 19 Levant morocco, calf lined, with flap edges 110 No. 5. Ruby 16mo. (6| X 4i X If inches.) Cloth boards, red edges . French morocco, gilt edges . Paste Grain morocco, limp . French morocco, circuit edges Best Turkey morocco, limp . Best Turkey morocco, circuit edges Levant morocco, lined calf, with flap edges No. 6. Pearl 16mo. (5| X 3| X 1^ inches.) Cloth boards, red edges . French morocco, gilt edges . Paste Grain morocco, limp French morocco, circuit edges Best Turkey morocco, limp , Best Turkey morocco, circuit edges Levant morocco, lined calf, with flap edges THE OXrOED BIBLE FOR TEACHERS CONTAINS THE FOLLOWINQ HELPS TO THE STUDY OF THE BIBLE. I. NOTES ANALYTICAL, CHRONOLOGICAL, HISTORICAL, GEOGRAPHICAL, ZOOLOGICAL, BOTANICAL, AND GEOLOGICAL. 1. Notes on the Old Testament:— i. Title of the Bible. ii. Hebrew Divisions of the Bible :— (a) The Law. (b) The Prophets. (c) The Scriptures. iii. Divisions of the English Bible:— (a) The Pentateuch. (b) The Historical Books. (c) The Poetical Books. {d) The Prophetical 13ooks. Analysis and Summary of each. 2. Summary of the Interval between the Old and New Testaments. 3. Family of the Herods. 4. Jewish Sects, Parties, &c. 5. Chronology of the Old Testament. 6. Chronology of the Acts and Epistles. 7. Historical Summary. 8. Miracles and Parables of the Old Tes- tament. 9. Miracles and Parables of Our Lord. 10. Names, Titles, and Offices of Christ. IL Prophecies relating to Christ. 12. Special Prayers found in Scripture. 13. Notes on the New Testament :— i. Early Copies. ii. Divisions of the New Testament:— (a) Constitutional and Historical (b) Didactic. (c) Prophetic. Analysis and Summary of each. 14. Harmony of the Gospels. 15. Paul's Missionary Journeys. 10. „ Voyage to Rome. 17. Geography and Topography of Pales- tine. 18. Mountains of Scripture, with their As- sociations. 19. Rivers and Lakes of Scripture, and Events connected with each. 20. Ethnology of Bible Lands. 21. Quadrupeds named in the Bible, with Description of each. 22. Summary of Mammalia op the Bible. 23. Fisheries of Palestine, with their Pro- ducts. 24. Aquatic Animals mentioned in the Bible. 25. Birds found in Palestine. 26. Reptiles of Scripture. 27. Insects op Palestine. 28. Trees, Plants, Flowers, &c., of Palestine. 29. Geology op Bible Lands :— L Mineral Substances, &c. ii. Metals. iii. Precious Stones. 30. Music and Musical Instbumeitts :— L Stringed Instruments, ii. Wind Instruments, iii. Instruments of Percussion. 31. Tables op Weights, Measures, Time, and Money. 32. The Jewish Year. 33. Words Obsolete or Ambiguous. 34. Words used Symbolically. 35. Blank Leaves for MS. Notes. II. AN INDEX TO THE HOLY BIBLE. III. THE NEW OXFORD CONCORDANCE. IV. DICTIONARY OF SCRIPTURE PROPER NAMES, WITH THEIR PRONUNCIATION, MEANINGS, AND REFERENCES. V. SCRIPTURE ATLAS (INDEXED). 1.— The Nations op the Ancient World. 2.— Armenia, Assyria, Babylonia, Syria, &c., IN THE Patriarchal Ages. 3.— Canaan in the Patriarchal Ages. 4.— Egypt and the Sinai Peninsula, illus- trating the Journeys of the Israelites to the Promised Land. 6.— Canaan as divided among the Tribes. 6.— Dominions of David and Solomon. 7.— The Kingdoms op Judah and Israel. 8.— Assyria and the Adjacent Lands, illus- trating the Captivities. 9.— Jerusalem and its Environs. 10.— Palestine in the Time of Our Saviour. 11.— The Roman Empire in the Apostolic Age. 12.— Map illustratinq the Travels op St. Paul. THE OXFORD BIBLE FOR TEACHERS, Cvtract^ from <©pinions{. "Tlie large collection of varied information which you have appended to the Oxford Bible for Teachees, in a form so readily available for reference, has evidently been com- piled with the greatest care ; and the testimony which you have received to its accuracy is a guarantee of its high value. I cannot doubt that the volume, in its various forms, will be of great service." — The Archbishop of Canterbury. "The notion of incliwding in one volume all the helps that a clergyman or teacher would be likely to want for the study of the Bible has never been realised before with the same success that you have attained in the Oxford Bible for Teachers. In the small edition (Ruby 16mo. thin), by the use of paper very skilfully adapted to the purpose, there is a Bible with an Atlas, a Concordance, an Index, and several Tractates on various points of Biblical antiquity, the whole, in a very solid binding, weighing a pound and an ounce : no great weight for what is really a miniature library. The clergy will probably give the preference to the larger book, marked No, 4, This includes the Apocrypha, with all the helps to the use of the Bible that distinguish the series. Its type is excellent. Many clergymen are obliged to write sermons when travelling from place to place. This volume would serve as a small library for that purpose, and not too large for the most moderate portmanteau. I think that this work in some of its forms should be in the hands of every teacher. The atlas is very clear and well printed. The explanatory work and the indices, so far as I have been able to examine them, are very carefully done. I am glad that my own University has, by the preparation of this series of books, taken a new step for the promotion of the careful study of the Word of God. That such will be the effect of the publication I cannot doubt." — The Archbishop of York. "It would be difficult, I think, to provide for Sunday-School Teachers, or indeed for other students of the Bible, so much valuable information in so convenient a form as is now comprised in the Oxford Bible for Teachers."— The Bishop of London. " The idea of a series of Bibles in different types, corresponding page for page with one another, is one which the Dean has long wished to see realised for the sake of those who find the type of their familiar copies no longer available .... The amount of information com- pressed into the comparatively few pages of the Appendix is wonderful. And the Dean is glad to hear that the help of such eminent contributors has been available for its com- pilation. The Concordance seems to be sufficiently full for reference to any text that may be required." — The Dean of Rochester, "Having by frequent use made myself acquainted with this edition of the Holy Scrip- tures, I have no hesitation in saying that it is a most valuable book, and that the ex- planatory matter collected in the various appendices cannot but prove most helpful, both to teachers and learners, in acquiring a more accurate and extensive knowledge of the Word of God." — The Bishop of Lichfield. "I have examined the Oxford Bible fob Teachers with very great care, and con- gratulate you upon the publication of so valuable a work. It contains within a reason- able compass a large mass of most useful information, arranged so conveniently as to be easily accessible, and its effect will be not merely to aid, but also, I think, to stimulate the studies of the reader. The book is also printed so beautifully, and is so handsome in every way, that I expect it will be greatly sought after, as a most acceptable present to any who are engaged in teaching in our Sunday Schools and elsewhere."— The Dean of Canterbury. THE OXFORD BIBLE FOR TEACHERS. (QytvacH ffom ©ptntonjEl {contmued)* '•'I have examined with some care a considerable portion of the * Helps to the Study of the Bible,' wliich are placed at the end of the Oxford Bible fob Teachers, and have been much stnick with the vast amount of really useful information which has there been brought together in a small compass, as well as the accuracy with which it has been com- piled. The botanical and geological notices, the account of the animals of Scripture, &c., seem to be excellent, and the maps are admirable. Altogether, the book cannot fail to be of service, not only to teachers, but to all who cannot afford a large library, or who have not time for much independent study." — The Dean of Peterborough, " I have been for some time well aware of the value of the Oxford Bible for Teach- ers, and have been in the habit of recommending it, not only to Sunday-School Teachers, but to more advanced students, on the groxmd of its containing a large mass of accurate and well-digested information, useful and in many cases indispensable to the thoughtful reader of Holy Scripture ; in fact, along with the Bible, a copious Index, and a Concor- dance complete enough for all ordinary purposes, this one volume includes a series of short but comprehensive chapters equivalent to a small library of Biblical works." — The Bishop OF Limerick. "Having examined the Oxford Bible for Teachers carefully, I am greatly pleased with it. Tlie 'Helps to the Study of the Bible' at the end contain a great amount of most valuable information, well calculated not only to lead to a good understanding of the text, but to stimulate the student to further efforts. It differs from many publications in this, that the information is so admirably arranged, that it is well suited for reference, and is easily available for the student. The edition would be most useful to Sunday-School Teachers, a great help to those who desire that the young shall have a real knowledge of the Word of God." — The Bishop of Cork, "The Oxford Bible fob Teachers may, I think, without exaggeration, be described as a wonderful edition of the Holy Scriptures. The clearness and beauty of the type, and the convenient shape of the volume, leave nothing to be desired. I know nothing of the same compass which can be compared to the 'Helps to the Study of the Bible' for fulness of information and general accuracy of treatment. It is only real learning which can ac- complish such a feat of compression." — The Bishop of Derby and Raphob. "I consider the Oxford Bible for Tbachers to be simply the most valuable edition of the English Bible ever presented to the public."— The Ven. Archdeacon Reichel. " The Oxford Bible for Teachers is in every respect, as regards type, paper, bindinor, and general information, the most perfect volume I have ever examined." — The Rev. Pre- bendary Wilson, of the National Society's Depository. " The essence of fifty expensive volumes, by men of sacred learning, is condensed into the pages of the Oxford Bible fob Teachebs."— The Rev. Andeew Thomson, D.D., Edinburgh. "The latest researches are laid under contribution, and the Bible Student is famished %oith the pith of them all." — Dr Stoughton. " The whole combine to form a Help of the greatest value."— Db. Angus, " I cannot imagine anything more complete or more helpful."— De. W. Moblby Ptjnshon. " I congratulate the teacher who possesses it, and knows how to turn its ' Helps ' to good account."— Dr. Kennedy. THE OXFORD BIBLE FOR TEACHERS. (Bytvacti from (©pim'onsf (continued). "The Oxford Bible for Teachers is the most valuable help to the study of the Holy- Scriptures, within a moderate compass, which I have ever met with. I shall make constant use of it ; and imagine that few who are occupied with, or interested in the close study of the Scriptures, will allow Such a companion to be far from their side." — The Rev. Baldwin Brotvn. "I do not think I shall ever leave home without the Oxford Bible for Teachers, for one can scarcely miss his ordinary books of reference when this Bible is at hand. I know no other edition which contains so much valuable help to the reader."— The Rev. A, H. Charteris, D.D,, Dean of the Chapel Royal. "The Oxford Bibles for Teachers are as good as ever we can expect to see." — The Rev. C. H. Spxjrgeon. "The modest title of the work scarcely does justice to the range of subjects which it comprehends, and the quality of their treatment. As a manual of Biblical information and an auxiliary of Biblical study, it is unrivalled. It is as exhaustive as it is concise,— no irrelevant matter has been introduced, and nothing essential to Biblical study seems to have been omitted,— and in no instance, so far as I can judge, has thoroughness or accuracy been sacrificed to the necessities of condensation." — The Rev. Robert N. Young, of Headingley College, Leeds. " The Oxford Bible for Teachers is really one of the greatest boons which in our day has been offered to the reading public. The information given is so various, and so com- plete, as scarcely to leave a single desideratum. To Christians, in their quiet researches at home, or in the course of extensive journeys, or in preparation for the duties of tuition, it is simply invaluable, and constitutes in itself a Biblical Library. The range of topics which it seeks to illustrate is very great, while the care and accuracy manifest in the articles desei'ves the highest praise. It is no exaggeration to say, that to the mass of Christian people it saves the expense of purchasing and the toil of consulting a library of volumes. At the same time, I know no book more likely to stimulate enquiry, and to give the power of appreciat- ing further research into the history, structures, and meaning of the Sacred Oracles." — Dr. Goold, of Edinburgh. "I have only recently possessed one of the Oxford Bibles for Teachers; and after a most patient examination of it, am astonished at the immense amount of accurate and carefully digested matter it contains : and that, too, of a kind precisely adapted to the Teacher's needs. "Would that such a Bible had been within my reach when I first began my teaching life ; and would that I had possessed one earlier, since I began to write for Teachers. As I look at its upwards of 300 pages of 'Helps to the Study of the Bible,' I recall my toilsome pilgrimage through many volumes, at much expenditure of time, for what is here so distinctly stated and tabulated. With such a desk companion I might have done so much more, and done it so much better. All I can do now is to entreat all Teachers who need a perfectly reliable Bible for study or class purposes, to procure one of the Oxford Bibles for Teachers. This I do most earnestly. And I would add that as a presentation volume by a class to its Teacher, or by a School to its Superintendent, no gift would be better appreciated or more appropriate than the small quarto size with its wide margin, and magnificent type, and superb flexible binding." — The Rev. James Comper Gray, Author of " Class and Desk," " Topics for Teachers,"' "Biblical Museum," ^c, ^c. "These admirable Bibles must tend to extend the fame even of the Oxford Press."— The Right Hon. W. E. Gladstone, M.P. THE OXFORD BIBLE FOR TEACHERS IS RECOMMENDED BY The AKCHBISHOP of CANTERBURY. The AKCHBISHOP of YORK. The BISHOP of LONDON. The BISHOP of WINCHESTER. ITie BISHOP of BANGOR. The BISHOP of CARLISLE. The BISHOP of CHICHESIER. The BISHOP of ELY. The BISHOP of GLOUCESTER and BRISTOL. The BISHOP of LICHFIELD. The BISHOP of LLANDAFF. The BISHOP of MANCHESTER. llie BISHOP of OXFORD. The BISHOP of PETERBOROUGH. TTie BISHOP of RIPON. The BISHOP of KOCH ESTER. The BISHOP of SALISBURY. ITie BISHOP of St. AI-BANS. The BISHOP of St. ASAPH. The BISHOP of St. DAVID'S. The BISHOP of WORCESTER. The BISHOP of SODOR and MAN. The BISHOP of BEDFORD. The DEAN of CANTERBURY. The DEAN of DURHAil. The DEAN of BANGOR. The DEAN of WELLS. The DEAN of ELY. The DEAN of EXETER. The DEAN of HEREFORD. The DEAN of LICHFIELD. The DEAN of LLANDAFF. Tae DEAN of MANCHESTER. The DEAN of NORWICH. The DEAN of PErERBOROUGH. The DEAN of RIPON. The DEAN of ROCHESTER. The Late DEAN of WORCESTER. CANON LIDDON. CANON GREGORY. The AKCHBISHOP of ARMAGH. The ARCHBISHOP of DUBLIN. The BISHOP of MEATH. The BISHOP of DOWN and CONNOR. The BISHOP of KILLALOE. The BISHOP of LIMERICK. The BISHOP of TUAM. The BISHOP of DERRY and RAPHOE. The BISHOP of CASHEL. The BISHOP of KILMORE. The BISHOP of CORK. The BISHOP cf OSSORY. The Vkn. ARCHDEACON REICHEL. The PRINCIPAL of the THEOLOGICAL COLLEGE, GLOUCESTER. The PRINCIPAL of the NATIONAL SOCIETY'S TRAINING COLLEGE, BATTERSEA. The CANO^ IN CHARGE of the DIVINITY SCHOOL, TKUKO. The PRINCIPAL of St. BEES COLLEGE. The PKTN<]IPaL of the THEOLOGICAL COLLEGE, WELLS. The PRINCIPAL of LICHFIELD THEOLOGICAL COLLEGE. The PRINCIPAL, St. DAVJD'S COLLEGE. The RIGHT HON. WILLIAM EWART GLADSTONE, M.P., LL.D. The Rev. A. H. CHAKTERIS, D.D., Profesxor cf Bibli- cal Criticism in the University of Edinbiiryh. Db. lee, Professor of Ecclesiastical Historf/ in the University of Glasgo^c. The RIGHT HON. JOHN INGLIS, D.C.L.. LL.D., Chancellor qf the UnivereUy cj/" Edinburgh, The EARL of SHAFTESBURY. Dr. ANGUS. Dr. SroUGHTON. The Rev. C. H. SPURGEON. Dit. RIGG, of the Westminster Normal Institution, Dr. KENNEDY. The Rev. EDWIN PAXTON HOOD. The Rev. W. MORLEY PUNSHON, LL.D. The Rev. HORATIUS BONAR, D.D. Dit. GOOLD, of Edinburgh. PROFESS .>R BINNIE, D.D. PROFESSOR BLAIKIE, D.D. Dr. ANDREW THOMSON, of Edinburgh. Dr. DAVID BROWN, Principal of Bee Church Col- lege, Aberdeen. PROFESSOR SALMOND, of Free Oiurch College, Aberdeen, Dr. W. LINI:SAY ALEXANDER. Dr. ALEXANDER MACLAREN. The Rev. PRINCIPAL RAINY, D.D., of New Col- lege, Edinburgh. Dr. JAMES MACGREGOR, of Edinburgh. Dr. ANTLIFF, Principal qf the Theological Insti- tute, Sunderla7id. Dr. NEWTH, of Neiv College. The Rev. E. E. JENKINS, M.A., President of the Wesleyan Conference. The Rev. M. C. OSBORN, Secretary of the Wesleyan Conference. Dr. GEORGE OSBORN, qf the Theological Institu- tion, liichmond. The Rev. R. GREEN. The Rev. W. HUDSON. I'he Rev. F. GREEVES. Dr. W. p. pope. Professor of Theology, Uidsbury. Dr. GERVASE SMITH. The Rev. GEORGE MARTIN. Dr. FALDING. Dr. CHARLES STANFORD. Dr. LANDELS. I'he Rev. JOHN H. GODWIN. The Rev. J. C. HARRISON. The Rev. JOSEPH WOOD, M.A. Dr. GUMMING. The Rev. COLIN CAMPBELL McKECHNIE. The Rev. R. TUCK, B.A. The Rev. PRINCIPAL McALL, of Haclney College. The Rev. ROBERT N. YOUNG, o//fea the Lord hatli commanded. 16 t And Moses diligently sough( the goat of the sin offering, and, be- hold, it was burnt : and he was angry with Eleazar and Ithamar, the sons of Aaron which were left alivej saying, CHAPTER XI. X fVhat beasts may^ 4 aiid what may not be eaten. 9 fV hat fishes. 13 What fowls. 29 The creeping thini;s which are unclean. wntcn are unciean. \ \ ND the Lord spake unto Mosej ' i ^ m .tl- and to Aaron, saving unto them, f-^ ' ^y*^. V 2 Speak unto the children of Israel, ., -^ # W 1 saying. These ore the beasts which ye >•>; O r- *^ m " The type of this dainty little volume, though necessarily vely minute, is clear and legible." — The Times. " It is printed on tough India paper of extreme thinness, and is wonderfully clear." — The Guardian. " When bound in limp morocco leather it weighs less than 3 J oz." — • The AthencBum. " It will pass through the post for a penny, and yet contains the whole of the Authorised Version from Genesis to E-evelation." — The Grai^hic. " It is not only a curiosity, but so convenient as not to encumber an ordinary waistcoat pocket." — The Globe. "It is smaller than an ordinary Prayer-Book, and can be read easily enough. * * * * It is a curiosity of the Binder's as well as of the Printer's art." — The Sjyectator. "Besides being a marvel as regards size, the little volume is a model of printing, highly creditable to the Oxford University Press." — llie Standard. LONDON : HENEY FEOWDE, OXFORD UNIVERSITY PRESS WAREHOUSE, 7, PATEENOSTEE ROW. if ^'^- .4/ /1