THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES / /^, i/i^cL, ^\Q^ '/>/"" PHYSICAL CELESTIAL MECHANICS. BY BENJAMIN PEIRCE, riiUKISS ritOFESSOR OF ASTRONOMY AND MATIIKM VTK S 1 \ IIAKVAIMI ISlVEIiSlTV, AND CONSULTING ASTRONOMER OF THE AMERICAN El'llLMEKlS AND NAUTICAL ALMANAC. DEVELOTED IN EOUIt SYSTEMS OF ANALYTIC MECHANICS, CELESTIAL MECHANICS, POTENTLVL PHYSICS, AND ANALYTIC MOPtPPIOLOGY. BOSTON: LITTLE, BROWN AND COMPANY. 1855. A S Y S T E Jil OP ANALYTIC MECHANICS. BENJAMIN PEIRCE PERKIXS PROFESSOR OF ASTR0X03IT AND MATHEMATICS IN HARVARD UNITERSITY, AND CONSULTING ASTRONOMER OF THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC. BOSTON: LITTLE, BROWN AND COMPANY. 18 5 5. Entered according to Act of Congress in the year 1855, by LITTLE, BROWN AND COMPANY, In the Clerk's Office of the District Court of the District of Massachusetts. CA:MlillIDGE: ALLEN AND F A K N 11 A M , P K I N T E R S . Engineering & Mathematical Sciences Library TO THE CUKRISIIF.D AND REVERED MEMORY OF MY :M ASTER IN SCIENCE, X A T 11 A N I E L B W D I T C II , THE FATHER OF A3IERICAN GEOMETRY, THIS VOLUME IS INSCRIBED. G21232 ADVERTISEMENT. The substance of the present volume was originally pre- pared as part of a course of lectures for the students of mathe- matics in Harvard College. But at the request of some of my pupils, and especially of my friend Mr. J. D. Runkle, I have been induced to undertake its publication. The liberality of my publishers, the well-known firm of Little, Brown & Co., who gen- erously gave directions to the printers, that no expense should be spared in its typographical execution, seemed to impose upon me an increased obligation to perform my portion of the task to the best of my ability. I have consequently reexamined the memoirs of the great geometers, and have striven to consoli- date their latest researches and their most exalted forms of thouo-ht into a consistent and uniform treatise. If I have, hereby, succeeded in opening to the students of my country a readier access to these choice jewels of intellect, if their bril- yjj.^ ADVERTISEMENT. liancy is not impaired in this attempt to reset them, if in their new constellation they illustrate each other and concen- trate a stronger light upon the names of their discoverers, and still more, if any gem Avhich I may have presumed to add, is not wholly lustreless in the collection, I shall feel that my work has not been in vain. The treatise is not, however, designed to be a mere compilation. The attempt has been made to carry back the fundamental principles of the science to a more profound and central origin ; and thence to shorten the path to the most fruitful forms of research. It has, moreover, been my chief object to develop the special forms of analysis, which are usually neglected, because they are only applicable to particular problems, and to restore them to their true place in the front ranks of scientific progress. The methods which, on account of their apparent generality, * have usually attracted the almost exclusive attention of the student, are, on the contrary, reestablished in their true position as higher forms of speciality. BENJAMIN PEIRCE. — 67 — shell is 1. For an internal point this equation becomes, by §§ 135 and 144, the intearral of which is /* M ' to which no constant need be added, because, when the dimensions of the shell are infinite, S2 and i2^ both vanish, since all the points of action are infinitely remote from the centres of action. This equation expresses that the potential of each shell has the same value for all internal points, and, therefore, there is no tendency to motion ^vithin the shell, and the surface of the shell must be level, with reference to its own action. 2. For an external point, the equation (G72) becomes, by §135, Hence, by integration, — = a constant, which constant, however, depends for its value upon the position of the points of action ; but since it has the same value for all the shells to which the point is external, the potential is constant for the same series of points external to one shell for which it is constant through the action of another shell ; that is, all the shells have the same external level surfaces. But the external level surface, which is nearest to any shell, differs infinitely little from — 68 — the level surface of the shell itself, and, therefore, the surface of each shell is a level surface for every included shell. Hence, the external level surfaces of a shell are the same with those of the original masses, and the attraction of a shell upon an external point has the same direction with the attraction of the original masses, and is normal to the level surface passing through the point. This theorem is due to Chasles. 146. Every infinitely thin shell, of ivhich the surface is level, from the action of the shell itself, must he a Chaslesian shell. For, if another shell is constructed upon this level surface, which is the negative of the Chaslesian, one, namely, which is repulsive, instead of being attrac- tive, or the reverse, and the whole mass of which is equal to that of the given shell, the two shells, having the same level surfaces, exactly cancel each other's action throughout all space. The elements of mass of the two shells must then be absolutely equal, but of opposite signs at every point. For, if they were unequal at any point, that point might be made the centre of an infinitely thin circular element of the combined shells. From the symmetry of its figure, a level surface for the action of this element alone might be made to pass through its perimeter, and which could inclose no other mass than the element itself But such surface cannot be level for the remainder of the combined mass of the two shells, and, therefore, the value of the potential upon this surface for the combined masses of both shells, including the circular element, cannot be constant. This want of constancy in the potential is contradicted by the fact that the shells balance each other's action everywhere. There cannot, therefore, be any such want of con- stancy, nor any point for which the element of mass of the given shell is not absolutely equal to that of the Chaslesian shell, although it is of a contrary sign. But reversal of the sign of the action of the mass does not interfere with the Chaslesian characteristic of the shell. — GO — 147. Two Chaslesian shells, v:Ucli are constructed upon the same surface, only differ in their densitf/ and their modulus of thickness. For the density of either of them may be increased or decreased mitil the value of its potential at the common surface shall be equal to that of the other shell. If, then, its action be reversed, the value of the potential for the combined shells will be zero both at the surface and at an infinite distance from the surface ; and it cannot have any other value in the intermediate space, otherwise, there would be points or surfaces of maximum potential exterior to the acting masses. The combined surfaces have, therefore, neither external nor internal action, and the reasoning of the preceding article demonstrates that the component shells are identical, except in reo:ard to their siorns. ATTRACTION OF AX ELLIPSOID. 148. An infinitelf/ thin homogeneous shell, of which the inner and Older surfaces are those of sinular, and similarly placed, concentric ellipsoids, is a Chaslesian shell. For, if upon the longest axes of these ellipsoids, as diameters, two concentric spheres are constructed, each sphere may be compressed into the corresponding ellipsoid, by reducing all the coordinates from the centre, as origin, parallel to either of the two shorter axes of the ellipsoid in the ratio of the longest axis to this shorter axis. But all points, which are originally in the same straight line remain upon a common straight line after this imiform compression ; and all distances which are measured in the same direction are reduced in a common ratio. But the thick- nesses of the spherical shell, measured upon any straight line at the two points where this line cuts the shell are equal ; so that the thicknesses of the ellipsoidal shell, measured at the two points where the reduced line cuts this shell, are also equal. If, then, at a — 70 — point assumed at will, as the vertex, within the ellipsoidal shell, an infinitesimal cone is constructed and extended in each direction from the vertex, till it intersects the shell, the relative masses of the two included portions of the shell are proportional to the squai^s of their distances from the vertex; and, therefore, their attractions upon the vertex are equal, but in opposite directions. Hence, the action of any portion of the shell upon an internal point is balanced by the action of the opposite portion, and there is, consequently, no tendency to motion within the shell from its own action. The surface of the shell is thus proved to be a level surface, in respect to its own action, and, by § 140, it can be no other than a Chaslesian shell. 149. This proposition may be enlarged to a theorem given by Newton, for a finite shell, of which the inner and outer surfaces are those of similar and similarly placed concentric ellipsoids. Such a shell may be called a Newtonian shell, so that the infinitely thin Newtonian shell is a Chaslesian ellipsoidal shell. But the New- tonian shell may be subdivided by similar and similarly placed concentric ellipsoidal surfaces into an infinite number of Chaslesian ellipsoidal shells, each of which is inactive with reference to an internal point. Hence, the tvhole Newtonian shell exerts no action wjion an internal imnt. 150. An ellipsoid may be converted into any other similar, and similarly placed, concentric ellipsoid by a process similar to that by wdiich the sphere in § 148 was changed to an ellipsoid ; that is, by increasing or decreasing the coordinates of each point, taken from the centre as origin, and parallel to either axis, in the ratio of the corresponding axes of the two ellipsoids. The points of the two ellipsoids, which correspond in this process, have been called by Ivory corresponding iwints. By this process, any Newtonian shell may be converted into another concentric and similarly placed LIST OF SUBSCRIBERS. J. I. Bowditch, (10 copies), Boston. John D, Rimkle, (5 copies), Cambridge. Chaimcey Wright, (2 copies). a C. H. Sprague, (2 copies). Maiden. W. C. Kerr, Davidson College, X. C George Eastwood, Saxonville. Charles PhilHps, Chapel Hill, X. C. Joseph W. Sprague, (2 copies). Rochester, N. Y. J. M. Chase, Cambridge. R. H. Chase, a Sharon Tyndale, a Isaac Bradford, a John Bartlett, (3 copies), a Gustavus Hay, Boston. F. J. Child, Cambridge. William C. Bond, for Observatory of Harvard College, (2 copies), a J. E. Oliver, (2 copies), Lvnn. C. W. Little, Cambridge. N. Hooper, Boston. C. F. Choate, (2 copies), Cambridge. X LIST OF SUBSCRIBERS. J. P. Cooke, Jr., B. A. Gould, Jr., Joseph Winlock, H. L. Eustis, Joseph Lovering, C. Gordon, Jared Sparks, A. Brown, WilHam G. Choate, J. F. Flagg, Jr., A. E. Agassi z, William G. Pearson, Charles Sanders, (2 copies), Theophilus Parsons, John Erving, Jr., Charles H. Mills, Edmund D wight, Edward Everett, J. H. C. Coffin, T. S. Hubbard, Mordecai Yarnall, James Major, James Ferguson, R. B. Hamilton, Stej^hen Alexander, (2 copies), James Walker, William Chauvenet, (4 copies), Washington Observatory, (5 copies), Thomas Hill, Waltham Rumford Institute, Charles Avery, Cambridge. (( a a u a a a Washington, D. C. Cambridge. a a a a Boston. a iC Annapolis, Md. Washington, D. 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Elias Loomis, New York City. John Tatlock, Williamstown College, Edward Pearce, Providence, R. I. James Mills Peirce. Cambridoje. C. W. Eliot, Xll LIST OF SUBSCRIBERS. F. W. Bardwell, Cambridge. Charles L. Fisher, a William Dearborn, Boston. Sidney Coolidge, a L. R. Gibbs, Charleston, S. C. J. D. Crehore, Newton. J. G. Cogswell, Astor Library, New York City. Wolcott Gibbs, a u Triibner & Co., (25 copies). London. William S. Haines, Providence, R. L Alexis Caswell, a « George M. Hunt, Stuyvesant, N. Y. Charles C. Snow, Brooklyn, N. Y. B. Westermann, New York City. W. H. Cathcot, Urbanna University, Ohio, S. C. Huntington, Pulaski, Orange Co., N. Y. Edward King, New York City. A. G. Harlow, Cambridge. C. A. Cutter, a R T. Paine, a C. F. Sanger, a John B. Tileston, i( J. M. SewaU, li B. S. Lyman, a C. H. Davies, Cl Isaac A. Hagar, a F. H. Smith, Universitv of Yiro-inia. ANALYTICAL TABLE OF CONTEXTS. CHAPTER T. M O T I O X , FORCE, A X D :M A T T K R . Section 1. The universality of motion, . 2. The spiritual origin of force. Page Section . 1 1 3. The inertia of matter, 1 Page . 1 CHAPTER II. MEASURE OF M O T I O X A X D FORCE. I. The JiIeasure of Motiox. 4. Uniform motion, .... 2 5. Velocitj- defined, .... 2 6. Formula of varying velocity (227), . 2 n. The Measure of Force. 7. Force and power, .... 2-3 8. The force is proportional to the ve- locitj-, 3 9. The mass defined, . 10. The inertia of a body, 3 3-4 III. Force of jiovixg Bodies. 11. The power is proportional to the square of the velocity, ... 4 12. The power is half the product of the force by the velocity, ... 4 13. Formula of varying force, . . 5 CHAPTER III. FUXDAMEXTAL PRIXCIPLES OF REST AXD M O T I O X. I. Tendency to Motiox*. 14. The total force of a system of bodies is the exact equivalent of the sum of its components, .... 5 1.5. The fundamental principle of equi- librium, ..... 5 16. The measure of the tendency to any form of motion, .... 5-6 , 1 7. Formula for the measure of the ten- dency to a proposed motion, ex- pressed by virtual velocities (''lo), 6-7 U. The Equations of Motion and Rest. 18. The equation of motion (813) and that of rest (8a,), 7-8 19. These equations can be decomposed into as many partial equations as there are independent elements, . 8-9 XIV ANALYTICAL TABLE OF CONTENTS. CHAPTER IV. ELEMENTS OF MOTION. I. Motion of Translation. 20. A point has three independent ele- ments of motion, from which any other elementary motion can be obtained by the formula (lO,:;), 9-10 21. Translation and rotation, being almost universal, are the most important forms of motion, . . . 10-11 22. Definition of translation, . . .11 23. Parallelojiiped of translation, . . 11 II. Motion of Rotation. 24. Definition of rotation, . . .12 25. Projection of rotation upon any axis, 12-13 26. 27. Parallelopijjed and parallelogram of rotations, .... 13-14 28, 29. Elementary rotations projected upon a given direction, . . 14-15 30, 31. Theorem of two systems of rectan- gular axes (15»g), . . . 15-16 III. Combined Motions of Rotation and Teanslation. 32. Decomposition of a rotation into a ro- tation about a parallel axis and a translation in a peri^endicular plane, 16 33. Combination of rotations about par- allel axes, 16 34. Equations of the axis of combined parallel rotations, . . . 16-17 35. Combined equal rotations about jjar- allel axes, 17 36. Combined rotations about opposite axes, 17-18 37. Combined rotations which are equiva- lent to a translation, . . .18 38. A couple of rotations, . . . 18 39. Combination of rotation and transla- tion, 18 40. Analysis of every possible motion of a solid, into a screw motion, . 18-19 41. The instantaneous axis of rotation, . 19 42. Special mode of conceiving the motion of a solid by means of the surfaces described by the instantaneous axis, . . . . . .19 43. Case in which the surfaces of §42 are developable, .... 19-20 44. Case in which the surfaces of § 42 are cylinders or cones, . . . 20-21 45. General case reduced to that of § 44, combined with translation, . 21 46. Axes of greatest curvature of the coni- cal surface, .... 21 47. Decomposition of the rotation into rotations about the axes of greatest curvature, . . . . .22 48. Relations of the velocities of rotation to that of the instantaneous axis, 22-23 49. Relations of the rotations when the surtaces are cylinders, . . .23 IV. Special Elements of Motion aj«'D Equations of Condition. 50. The independent elements of position, 24 51. The equation of condition for depend- ent elements, .... 24-25 52. 53. Elimination by the method of mul- tipliers, 25-26 54. The variation of the equation of con- dition expressed by means of the variation of the normal to a cor- responding surface, . . . 26-27 ANALYTICAL TABLE OF CONTENTS. XV CHAPTER V. F O 1{ C E S OF NATURE. I. The Potential, Level Surfaces, Posi- tions OF Equilihrium, and the Pos- sibility OF Perpetual Motion. 55. The fixed laws are not incompatible with the spiritual origin of force, . 28 56. Fixed and variable forces, . . 28 57. The relation of the forces of nature to form expressed by the potential, 28-29 58. The dependence of the power of a system upon its form, . . .29 59. Limits of motion of a system, . . 29 60. 61. The potential is a maximum or a minimum for the position of equi- librium, 29-30 62. The relation of stability of equilibrium to the maximum or minimum of the potential, .... 63. There are as many positions of stable as of unstable equilibrium, with ref- erence to each element of equi- librium, ...... 64. The necessity of the potential in the fixed forces of nature, and its rela- tion to the possibility of perpetual motion, ...... 65. The level surface and its finite extent, 66. The direction of attraction is perj^en- dicular to the level surface, . 67. The law of attraction determined by the distance apart of two infinitely near level surfaces. Level surfaces do not intersect each other. The ' continuity of the potential of nature, 32 68. The trajectory of level surfaces termi- nates in a maximum or minimum, 32-33 69. The limits in space of the constant potential coincide with those of the discontinuity of the potential or Its derivatives, 33 70. There is no force or mass throughout a space of constant potential, . 33 71. The potential of nature and its deriva- 30 30 32 33 tlves are finite and continuous throughout a space which contains no mass, ..... 72. A j^ortlon of sjiace, for which the po- tential of the fixed forces of nature is constant, is completely bounded by a continuous mass, . . 33-34 73. The potential of nature for tempora- rily fixed forces may vanish for an infinite extent of space, 74. 75. The computation of the difference of the potential for two points by the formula (342rj), 34 34 n. Composition and Resolution of Forces. 76. 35 35 All the phenomena of nature dejiend upon combined forces, . The projection of a force in a given direction, ..... The action of a combination of forces in any direction, . . . 35-36 The par aUelopiped of forces^ . .36 8L The resultant of forces, and its al- gebraic expression (3 705), . . 36-37 The tendency of a system to a motion of translation, .... 37-38 The moment of a force, . . .38 The jjrojectlon of a moment, . . 38-39 The par allelopiped of moments, . The moment of a force measures its tendency to produce rotation. The positive direction of the axis of a moment, ..... The resultant moment measures the total tendency to produce rotation. The resultant moment of forces which act upon a point is the moment of their resultant, .... 9L The moments for parallel lines, The resultant moment ibr different points, ...... 39 39 39 40 XVI ANALYTICAL TABLE OF CONTENTS. 93. A couple of forces, ... 40 94. The moment of a couple is constant for all points of space, . . .40 95. The tendency of a system of forces to produce translation and i-otation may be reduced to a resultant and a resultant couple, . . . 40-41 96. It may be still further reduced to two forces, . . . . .41 97. The resultant and the resultant moment may always coincide in direction, . . . . .41 98. 99. If the forces arc in the same plane, or if they are parallel, the combined equivalent is either a resultant or a resultant moment, . . .42 100. Analytic determination of the com- mon direction of the resultant and resultant moment (489), . . 42-43 101. The special reduction of forces re- quires special forms of analysis, . 43 III. Gravitatiox, and the Force of Statical Electbicity. 102. Gravitation, and Its elementary po- tential, 43 103. Statical electricity, and its element- ary potential, .... 43-44 104. Law of distribution of electricity, . 44 105. Potential of gravitation and electric- ity (45g), 44-45 106. Laplace's equation for the determi- nation of the potential (465), . 45-46 107. The law of attraction of gravitation or electricity (40,;), . . .40 108-112. The Attraction of an Infi- nite Lamina, .... 46-48 108. The potentlalof an infinite lamina, 46-47 1 09. The level surtaces of a uniform lamina, 47 110-112. The attraction of a uniform la- mina (484) and (48io), . . 47-48 113. Poisson's Modification of La- place's Equation for an In- terior Point (49o), . . 48-49 114-124. The Attraction of an In- finite Cylinder, . . . 49-54 114. The form of the potential of an in- finite cylinder (5O3,), . . . 49-50 115. The level surfaces of an infinite cyl- inder, 50 116. Form of the attraction of an infinite cylinder (oO^.i^), . . . .50 117-120. The attraction of an infinite cyl- inder upon a distant point (51 29), (52i0, 50-52 121-123. The attraction of a circular cyl- inder (53^,)' (542), (54;), (54,,),' 52-54 125, 126. Relation of the Poten- tial to its Parameter, . 54-55 127, 128. Attraction of a Finite Point upon a Distant Mass. The Centre of Gravity, . 55-50 129-132. The Attraction of a Spher- ical Shell (56^,), (57,9), (5728), (58s), 56-58 133-147. The Action and Reaction OF A Surface or infinitely THIN Shell of finite extent. The Ciiaslesian Shell, . 58-69 133. The total action of a surface normal to itself, 58-59 134. That of a plane, . . . .59 135. Gauss's theorem relative to the angle subtended by a surface (60^4), . 59-60 136. Gauss's and Chasles's 225 425. Case in which the given equations cannot be reduced to the normal form without differentiation, . . 225 42G. Direct determination of the functions involved in the equation of the mul- tiplier from the given equations, 225-226 427. Determination of the factor for the passage from the multiplier of the given equation to that of one of the simplest forms of normal equa- tions (2288), 226-228 428-43L Principle of the Last Mul- tiplier, .... 228-231 428. The Jacobian multiplier coincides with the Eulerian multiplier when there are two variables (229o), 228-229 429. Jacobi's princij^le of the last mul- plier, 229 430. By the principle of the last multiplier, when the element of imriation is 7iot directly expressed in the given equa- tions, either the tico last integrals can he obtaitied by quadratures, or the last integral can he obtained iviih- out integration, .... 229 431. The principle of the last multiplier when a portion of the variables are not involved in the remaining de- rivatives, and the remaining de- rivatives satisfy a given equation, 230-231 432-441. Partial Multipliers, . 231-235 432. Definition of the partial multiijliers (231i0, 231 d 433, 434. Defining equation of the partial multipHer (231o8), (232j), . 231-232 435. Determination of the signs in the for- mation of the multipliers (232,:,), . 232 436. Case in which the partial multiplier is the Jacobian multiplier, . . 232 437. Case in which the partial multijilier is the Eulerian multiplier amplified by Lagrange, .... 232 438. Every partial multiplier corresponds to an integral of the equation, . . 233 439. The deduction of an integral from the Eulerian multiplier (23807), 233-234 440. Transformation of the partial multi- plier when there is a change of va- riables (239.,r) .... 239 441. Transformation and reduction of the partial multiplier -ndien the solutions are adojsted as new vari- ables, .... 234, 235 III. Integrals of the Differential Equa- tions OF Motion. 442. General integrals of the equations of motion, 235 443-451. The Application of Ja- cobi's Principle of the Last Multiplier to Lagrange's Canonical Forms, . 236-241 443. Lagrange's canonical forms consti- tute a system of normal forms 236 444. A Jacobian multiplier is always known in equations of motion when the forces do not involve the veloc- ities (237i8), . . • . 236-237 445. The principle of the last multiplier ex- pressed as a dynamiccd principle, . 237 446. The Jacobian multiplier ichen the equations of motion are expressed in rectangular coordinants (23813), 237-238 447. The conditional equations expressed in the multiplier of the ecpiations of motion (239,,,), . . . 238-239 448. The transformation of the multiplier hy the introduction of the original XXVI ANALYTICAL TABLE OF CONTENTS. elements instead of the rectangular coordinates (44O3), . . . 239-240 449, The Jacobian multiplier of the equa- tions of motion when there are no equations of condition ; it is unity when the coordinates are rectangu- lar (240i,), 240 450. The equations of condition considered as forces in the expression of the multiplier (241;), . . . 240-241 451. The multiplier is unity when the dif- ferential equations of motion are expressed In Hamilton's form, . 241 CHAPTER XI. MOTION OF TRANSLATION, 452. The motion of the centre of gravity is Independent of the mutual con- nections, .... 241-242 453. The motion of the centre of gravity depends upon the external forces, 242 454-752. Motion of a Point, . 242-433 454. The ditierentlal equations of the mo- tion of a point (2434), . . 242-243 455-459. A Point moving upon a Fixed Line, . . . 243-244 455. By the principle of the multiplier, the motion is expressed hij integrals by quadratures (243ig, 24); • • • 243 456. The velocity dependent solely upon position, and not upon the interme- diate path, .... 243-244 457. Case in which the motion Is limited, in which case the oscillations are Invariable in duration, . . . 244 458. If the path returns into itself, the jieriod of circuit is constant, . . 244 459. Exjiression of the multiplier when the forces and equations of motion involve the time (24428), ■ • 244 460-477. The Motion OF a Body UPON •A Line v^^hen there is no Ex- ternal Force. Centrifugal Force, .... 245-254 460. Upon a fixed line with no external force the velocity is constant, . 245 461. Measure of the centrifugal force (245i,), 245 462. Total pressure upon a line where there are external forces, . . 245 463. The centrifugal force cannot be used as a motive power, . . . 245 464. The acceleration of a body upon a moving line (24620), . . 245-247 465. Upon a uniformly moving line the relative velocity of a body acted upon by no force is constant, . 247 466. The acceleration of a line moving with translation is a negative force acting upon the body, . . .247 467. The same proposition applies to any line, 247-248 468. Case in which the line rotates uni- formly about a fixed axis (24804), • 248 469. The time of oscillation of a body up- on a uniformly rotating line Is constant, .... 248-249 470. The period of circuit of a body upon a uniformly rotating line is constant, 249 471. Case in which the motion of the body vanishes at the axis of rotation (2492,,), 249 472. Motion of a body on a uniformly rotating straight line (250g2o, 31)5 (25I2), 249-251 473. Motion of a body on a uniformly ro- tating circumference of which the plane Is perjjendlcular to the axis of rotation (25I29), (2523_ 22,20)? (2533,11,21), .... 251-253 474. Motion of a body upon a rotating line which is wholly contained up- ANALYTICAL TABLE OF CONTENTS. XXVU on the surface of a cylinder of revo- lution of which the axis is the axis of rotation (25333), • • . 253 491 475. Case in which the rotation of the cyl- inder is uniform, .... 254 476. Case in which the curve is a helix (254g), 254 477. Case in which the acceleration is uniform (254i5), .... 254 478-482. MoTiox OF A Heavy Body UPON A FIXED Line. The Sim- PLK Pendulum, . . . 254-256 478. The motion of a heavy body upon a fixed line (25428), .... 254 479. When the line is contained upon the surface of a vertical cylinder, 254-255 480. When the line is straight (255^), . 255 481. WTien the line is straight and no ini- tial velocity (2551.,), . . . 255 482. When the line is the circumference of a circle ; oscillations of the sim- ple pendulum (255o5), (2563,16,530,25), 255-256 483-502. Motion of a Heavy Body UPON A moving Line, . 257-270 483. When the line has a motion of trans- lation (25 7j), .... 484. When the translation is uniformly accelerated, it is equal to a constant force, ...... 485. When the line is straight, and the law of translation is given ; in what case this path Is a parabola (257^.31), (258,, n), . .257- 486. When the translation of the line is uniform and direct ; gain of power (258,0, (259i6-25), • . • 258-259 487. When the line is the circumference of a vertical circle (2593i), (26019,24), (26 lo, 8, 14.18), .... 259-261 488. When the line rotates about a verti- cal axis (26I03), . . . .261 489. When the line rotates uniformly about a vertical axis (262.), . 261-262 490. When a straight line rotates uni- 257 257 -258 492. 493. 494. 495. 496. 497. 498. 499. 500. 501. 502. 263 264 formly about the vertical axis (262i3), 262 Direct integration of the linear dif- ferential equation in this case Into the form given by Vieille (262io), 262 Case of § 490, In which there is an impassable limit (26205,31), (263„), 262-263 Case of § 490, in which there is no hmit (263ji^), Case of § 490, in which there Is a possible position of immobility (264i.io), .... When the circumference of a circle rotates uniformly about a vertical axis ; the point of maximum and minimum velocity defined by an hyperbola (265io), . . . 264-265 Case of § 495, In which there is no motion upon the line, . . . 266 Case of- § 495, when the minimum velocity vanishes (26 7,,i), . 266-267 \^Tien a parabola of a vertical trans- verse axis rotates uniformly about its axis (26810, 31), . . .267-269 Case of §498, when the minimum velocity v^anishes (269ii), When the axis of rotation Is not ver- tical, and when the rotation is uni- form (269is), .... When a straight line rotates imlform- ly about an inclined axis (270^), 269-270 Rotation of a plane curve about an Inclined axis (270i:j), . . . 270 269 269 503-534. Motion of a Body upon a Line in opposition to Fric- tion, OR through a resisting Medium, .... 270-315 503. The resistance of a medium, . 504. Expression of the resistance, 505. Resistance of a medium to the mo- tion of a body upon a fixed line (2713,9), . . . . . 506. Motion of body upon a fixed line through a resisting medium with- out external force (271i2, k), . 270 270 271 271 XXVlll ANALYTICAL TABLE OF CONTENTS. 507. 508. 509. 510. 511. 512. 513. 514. 515. 516- 51G. 517, 518, 519, 520, 521 Case of § 506, when the law of resist- ance is expressed as a quadratic function of the velocity. Change of sign of the resistance (^271o^), (2724,13.22), .... 271-273 Case of § 506, when the resistance is friction upon the line (273i9), . 273 Case of § 508, Avhen thei"e is no ex- ternal force (273os), . . .273 Case of § 509, when the fixed line is the involute of the circle (274^), . 274 Case of § 509, when the line is the logarithmic spiral (274i4), . . 274 Case of § 509, when the line is the cycloid (274._>.,), . . . .274 Case of § 506, when the resistance of the line is constant, and the resist- ing medium moves with a uniform velocity, and the resistance is pro- portional to the velocity (2753), 274-275 Case of §513, when the line is straight, and there is no external force (275„_2o), . . . 275-276 Motion of a heavy body upon a fixed straight line, when the resistances are friction, and that of a moving medium which resists as the square of the velocity (27631), (2773,29), (278i6), (2793,15,28), (280io,2o), 276-280 -534. The Simple Pendulum in A RESISTING MeDIUM, . 281-315 The small oscillations of a pendulum against friction and the resistance of a medium which is proportional to the velocity (28I20.20), . . 281 The oscillation after many vibra- tions (2824), . . . • .282 The time of oscillation compared with that in a vacuum (282ig,23), . 282 The arc of oscillation (283.,), . 282-283 The law of diminution of the arc of oscillation and of the maximum of velocity (283.,8), . . . .283 The oscillations of the pendulum if the resistance is as the square of the velocity (28403, 31), • • 284-285 522. The arc of oscillation in the case of § 521 (2852«) 285 523. The arc of oscillation in the case of § 521 is the same as in a vacuum (28612,17,0.2), .... 285-286 524. The oscillations of the pendulum when the law of resistance is ex- pressed as a function of the time (287„g), .... 286-287 525. The oscillations of the pendulum, as affected by those produced in the medium (28802,04), (28924_oo), (290.2„), (2914), . . . 287-291 526. The oscillations of the pendulum as affected by the portion of the me- dium which becomes part of the pendulum (291o8_3i), • . 291-292 527. Constants of the formulae of the oscil- lations of the pendulum in a resist- ing medium arranged for applica- tion to experiment (29 2i:_i9), . . 292 528. Approximate form for the best exper- iments in which the friction is in- sensible (29224_28), .... 292 529. The French system of weights and measures adopted in the examina- tion of experiments, . . 292-293 530. Discussion of Newton's experi- ments upon the pendulum in air (293ig.2o, 26-28), .... 293-294 531. Discussion of Dubuat's experi- ments upon the pendulum in air and water (2956.7, iwe), • . 294-295 532. Discussion of Borda's experi- ments upon the pendulum in air (2965^,ii_,3,n.i8), . . . 296-297 533. Discussion of Bessel's experi- ments upon the pendulum in air, (2987_9, i3_i5, 20-22, 25-2;), (299ii_i3, 15.17), (2990,^22,25-27),. • • . 298-311 534. Discussion of Baily's experiments upon the pendulum in air, . 311-315 535-559. The Tautochrone, . 316-327 535. Definition of the tautochrone, . .315 536. The case of the tangential force of the tautochrone when it can be ex- ANALYTICAL TABLE OF CONTENTS. XXIX pressed as a function of the arc (3179), 316-317 537. The equation of the tautochrone un- der the action of a fixed force (317ie), 317 538. The tautochrone which rotates uni- formly about a fixed axis when there is no external force (31 73.5), . 317 539. The case of § 538, when it is a plane curve (3184.7), . . • 317-318 540. The cycloid is the tautochrone of a free heavy body in a vacuum (31823) 318-319 541. The tautochrone of a heavy body in a vacuum upon a given surface (319i3), 319 542. The tautochrone of §541, when the surface is a cylinder of which the axis is horizontal, and the equa- tion of the base is (31905), (320;), 319-320 543. The tautochrone of § 542 upon the developed cylinder (320,4), . . 320 544. The tautochrone of §542, when it passes through the lowest side of the cylinder (320i4.,„,.>4), • 320-321 545. The differential equation of the tau- tochrone of §542 referred to rec- tangular coordinates (32I10.15), . 321 546. The tautochrone of § 542, when the base of the cylinder is a cycloid (32I2.,), (322^), . . . 321-322 547. The tautochrone of a heavy body upon a surface of revolution of which the axis is vertical, and the meridian curve is that of (31 905), (322,,), 322 548. The tautochrone of a heavy body upon a vertical cone of revolution (322^8), (323,), . . . 322-323 549. The tautochrone of § 548, which passes through the vertex (3239), . 323 550. The tautochrone of § 547, when the meridian curve is a cycloid (323,0, 323 551. The tautochrone upon a plane when the force is directed towards a point in the plane, and propor- tional to some power of the dis- tance from the point (323,.-,), . 323 552. The tautochrone of § 551, when the force is any function of the dis- tance (324,,;), .... 323-324 553. The polar differential equation of the tautochrone in the case of § 552 (324^0, 324 554. The difl[erential er|uation of the tau- tochrone of § 552 in terms of the radius of curvature and the angle of direction (324o7), • • . 324 555. The tautochrone of §552, when it is the involute of the circle (325,,), . 325 556. The tautochrone of §552, when it is a logarithmic spiral (32526), 325-326 557. The tautochrone of §552, when the force is proportional to the dis- tance from the origin ; when it is not infinite, it is an epicycloid (32G2,), (327,o), . . . 326-327 558. Cases included in § 557, near the point of greatest velocity, . .327 559. The tautochrone in a resisting me- dium postponed to case of holo- chrone, 327 560-604. The Brachistochroxe, 328-354 560. Definition of the brachistochrone, . 328 561. The investigation of the free brachis- tochrone (328,9), • • • .328 562. The brachistochrone when the act- ing forces are fixed (328,4, js), • 328 563. The pressure upon the brachisto- chrone is double the centrifugal force (3294), .... 328-329 564. The point of contrary flexure in a brachistochrone, . . . .329 565. The conditions of the brachisto- chrone introduced by the general method of variations, . . . 329 566. When the force is directed towards a point, the free brachistochrone is a plane curve, and its plane in- cludes the point of attraction, . 329 567. When the forces are j)ai'allcl, the free brachistochrone is a j)lane XXX ANALYTICAL TABLE OF CONTENTS. curve, and its plane is parallel to the direction of the forces, . .329 568. When there are no forces the brachis- tochrone is the shortest line, . 329-330 569. The equation of the brachistochrone Avhen its force is central (330^-,), . 330 570. The brachistochrone of § 569, when the force is proportional to the dis- tance ; it is a spiral or an epicy- cloid (331io, 5„), . . . 330-331 571. The equation of the brachistochrone when the forces are parallel (33l3j), 331-332 572. The brachistochrone of a heavy body is a cycloid (332o), . . .332 573. The brachistochrone of § 571, when the force is proportional to the dis- tance from a given line (332io), (333io), .... 332-333 574. The centrifugal force in the brachis- tochrone upon a given surface, 575. Simple case of a brachistochrone upon a given surface, including that of a meridian line upon a sur- face of revolution, . . 333- 576. The brachistochrone upon a surface of revolution when the force is di- rected to a point of the axis, and expression of the projection of the area upon the plane perpendicular to the axis (334,g_.,o), . . 334-335 577. The derivatives of the arc and of the difference of longitude in the case of § 576, taken with reference to the arc of the meridian (3355.7), 578. The derivatives of the same quanti- ties taken with reference to the latitude (331ii_i,j), .... 579. The surface upon which the brachis- tochrone may make a constant angle with the meridian ; it may be used to define the limits of the brachistochrone in any case of § 576 (335,„), (336o), . . 335-336 580. The limiting surface of § 579 is a paraboloid of revolution in the case of a heavy body, of which the axis is directed downwards. Investiga- 333 -334 335 335 581 582. 583, 584. 586. 587. 589. 590. 591. tion of the other brachistochrones upon this surface (33629), (33 73 j^^ „;), (3382. 7^ i3_ ir|_oo), (3395, 13^ oe)) (3407_ 14), 336-340 The brachistochrone for a heavy body upon a paraboloid of revolu- tion of which the axis is the upwai'd vertical (34O21), (341s), • • 340-341 The brachistochrone of a heavy body upon a vertical right cone (341 20), (3422.5, 10. 23), (3432, 13, 10, 22. 25), (3442), 341-344 The brachistochrone of a heavy body upon an ellipsoid of revolution of which the axis is vertical (344i4), (345io, 05), (3460.6, j„9, 25),. . 344-346 The tangential radius of curvature of the brachistochrone of a heavy body upon any surface (3477), . 346-347 When the force is parallel to the axis, and proportional to the distance from a plane which is perpendicular to the axis, the limiting surface of § 579 is an ellipsoid or an hyper- boloid, ..... When the force is proportional to the distance in § 576, the limiting sur- face of § 579 is an ellipsoid or an hyperboloid, Investigation of the limiting surface of § 579 when the force is propor- tional to the square of the distance in § 576, .... The normal pressure upon the brach- istochrone when the length of the arc is given (347._h,), . .347- The equation of the brachistochrone in the case of § 569, when the length of the arc is given (348,5,10), . The equation of the brachistochrone in the case of parallel forces when the length of the arc is given (348h,:s), .... The equation of the brachistochrone in the case of § 576, when the length of the arc is given. The investigation of the limiting sur- face (34805,09), (3495), • • 348-349 347 347 347 -348 348 348 ANALYTICAL TABLE OF CONTENTS. XXXI 349 349 592. The normal pressure in a brachisto- chrone when the total expenditure of action is given (349io), 593. The equation of the brachistoohrone in the case of § 576, when the to- tal expenditure of action is given (34924,27), 594. The equation of the brachistochrone in the case of parallel forces -when the total expenditure of action is given (34931), (350.,), . . 349-350 595. The equation of the brachistochrone in the case of § 576, when the to- tal expenditure of action is given, and the investigation of the limit- ing surface (350s_ 10,19), . . . 350 596. The brachistochrone in a medium of constant resistance (35l3_g, 13), 350-351 697. The expression of the multiplier of the equation for the element of length of the arc when the force is central in the case of § 596 (35I20), 351 598. The expression of this multiplier when the forces are parallel (35I24), 351 599. The ecpiation of the brachistochrone of a heavy body in a medium of constant resistance (35I29), . . 351 The brachistochrone in a medium, of which the resistance is a given function of the velocity (352]o^ig 05), 352 The equations of § 600 when the forces are parallel (35209^1), . . 352 602. The brachistochrone of a heavy body in any resisting medium, and in the case of the resist (3823,6), 380-382 680. Case of § 678 for a force of four terms which are inversely i)ropor- tional to the second, third, fourth, and fifth powers of the distance (382i5, 19, 24, 27, 3l)) .... 382 681. Case of § 678 for a binomial form of the radical of (37800), (3885, «), . 383 682. The general forms of force of § 676 which admit of simple integration consist of two terms, of which one is inversely proportional to the cube of the distance, and the other is proportional to the distance or inversely proportional to the square of the distance (383i3), . . .383 683. The term, wJiich is inversely propor- tional to the cube of the distance, does not increase the difficulty of integra- tion, and the effect of this term may be disguised in the constants (3833,), 383-384 684. Case of no force and of a central force inversely proportional to the cube of the distance (384,o,2i,27)» (3853, 6, K,), .... 384-385 685. Case of a central force proportional to the distance, . . . 385-386 686. Case of § 685 combined with a force inversely proportional to the cube of the distance (38 6i5_ig), . .386 687. Case of a central force inversely jjro- jjortional to the square of the dis- tance, 386-387 688. Case of § 687 combined with that of § 684 (387,^28), (388i„_„), . 387-388 689. Case of § 683 Avith the force of § 684 (388og), . . . .388 690. The general laws of force for which integration may be effected by el- liptic integrals, each consisting of four terms, with a total variety of six cases (3884,8), . • • 388-389 691. Second case of the first form of § 690 when the force consists of terms of the form of § 679 (390o„^,), (391i2-„), (392o,),(393«, 19-23), (3 94^„), (394,9.04), (396,0, (397,.;), (3972,.24), (3983,2(^23), (399,_9,i2), . . 389-399 692. First case of the first form of § 690, when the four terms are respec- tively projiortional to the distance, to its third and fifth powers, and to its inverse cube (399i5, 19, 22-31), (400,_,8), .... 399-400 693. Case of § 692, in which the force is proportional to the fifth power of the distance, 400 694. Case of §692, in which the force is , proportional to the cube of the dis- tance, 401 695. Third case of the first form of § 690, in which the four terms are in- versely proportional to the cube root of the distance, to the fifth and seventh powers of the cube root, and to the cube of the distance (401,5, ,8, 25), 401 696. Case of § 695, in which the force is inversely proportional to the cube root of the distance, . . 401-402 697. Case of §695, in which the force is ANALYTICAL TABLE OF CONTENTS. XXXV inversely proportional to the fifth and the seventh powers of the cube root of the distance, . . . 402 698. Fourth case of the first form of § 690, in which the four terms are inverse- ly proportional to the square and cube of the distance, and the third and fifth powers of the square root (402.,;. 09), (403,), . . . 402-403 699. Cases of §698, in which the force is inversely proportional to the third and fifth powers of the square root of the distance, .... 403 700. First case of the second form of § 690, in which the four terms are inverse- ly proportional to the second, third, fourth, and fifth powers of the dis- tance (403.,,), (4043,5), . . 403-404 Case of § 700, in which the force is inversely proportional to the fourth power of the distance, . . . 404 Case of § 700, in which the force is inversely proportional to the fifth power of the distance, . . 404-405 Second case of the second form of § 690, in which the four terms are proportional to the distance, and inversely proportional to the third, fifth, and seventh powers of the distance (40014,16,03), . . . 405 704. Case of § 703, in which the force is inversely proportional to the sev- enth power of the distance, . 405-406 705. Third form of central tbrce, in which the integration can be performed by elliptic integrals (406^,27,31)5 • 406 706. The potential curve for defining the limits of the path described under the action of a central force (407io), 407 707. The term of the jjotential, which cor- responds to the force of § 686, may be omitted in the potential curve of §706, 407 708. Potential curve in which the path can only consist of a single portion, 407—408 709. The portion of the potential curve which corresponds to attraction and repulsion, ..... 408 701 702 703 710. 711. 712. 713. 714. 715. 716. 717. 718. 719. 720. 721. 723 724 Form of the path for a central force in the vicinity of the centre of ac- tion, 408-409 Character of the path for a central force at an infinite distance from the centre of action, . . . 409 Graphic determination of the incli- nation of the path to the radius vector, 409 The equation of the path for parallel forces (4IO9), . . . .410 The path of a projectile is a para- bola (41O18), 410 The equation of the curve for par- allel forces referred to rectangular coordinates (41O31), . . .410 The potential curve for parallel forces, 411 Case in which the force of § 713 is proportional to the distance from a fixed line (411 i2_i5), . . . 411 Case in which the force of § 713 is proportional to the distance from any fixed line divided by the square of the distance from an- other line (41L,,;), . . . .411 The motion of a body upon a surface of revolution when the force is cen- tral, and the centre of action is upon the axis of revolution (4123^), 411-412 Derivatives of the arc and of the longitude in the case of §719 (412„.3,i5.n), 412 Case of §719, in which the path of the body makes a constant angle with the meridian. The surface of revolution which defines the limits ofthe path (41 2oi), (4135), . 412-413 Limiting surface of revolution for a heavy body (413,0, • • -413 Motion of a heavy body upon a verti- cal right cone (4 1 43_ii, 22-31) , (4 1 5 j_n) , (415is.25,3i), (4I63),. . .413-416 Motion of a heavy body upon a verti- cal paraboloid of revolution of which the axis is directed downwards (416i,_.,3, ,^.31), (4172,11.1,0, • 416-417 XXXVl ANALYTICAL TABLE OF CONTENTS. 725. Motion of a heavy body upon a ver- tical paraboloid of revolution of ■vvhicli the axis is directed upwards (4172.^), 417 726-735. The Spherical Pendulum, 418-423 726. The path of a spherical penduliun. (418,8, ,0, 418 727. Relation of the limits of the path of the pendulum (41S,-^.,), . .418 728. The time of oscillation for different lengths of pendulums, . . 418-419 729. Case in which the path of the pendu- lum is a horizontal circle (41 On, 17,23)) (4208, „), . . . . 419-420 730. The time of a complete revolution in the case of § 729 (420^4), (421^, 420-421 731. The path of the pendulum when it is nearly a horizontal circle (421,0.18), ' • • • • . .• 421 732. The path of the pendulum when it is nearly a great circle (42109,31), 421-422 733. The path of the pendulum when it passes nearly through the lowest point of the sphere (422io, 12), . 422 734. Limits of the arc of vibration of the pendulum, ..... 422 735. The azimuth of the pendulum (4232o, 25), 422-423 736-752. The Motion of a Free Point, 424-433 736. The acceleration of a free point in any direction (4246), • • .424 737. The rotation-area with reference to the moment of the force about an axis (4242g), .... 424-425 738. The potential for a central force pro- 425 425 portional to the distance. The path is a conic section (425i2), 739. The area for forces directed towards a line is proportional to the time, 740. The path for the case of forces di- rected towards a line investigated by means of the peculiar coordi- nates of the distances from two fixed points of the line (426i,, 31), (4273), (428i„.,9), . . . 425-428 741. The special cases of §740 may be combined into one by addition, . 428 742. Cases in which the forms of § 740 are expressed by elliptic integrals (42807.09), .... 428-429 743. Case of § 740, in which there are two forces which follow the law of gravitation, .... 744. Case of § 740, in which there is one force proportional to the dis- tance, ..... 745. Case of § 740, in which there is one force inversely proportional to the distance from the fixed line, . 429-430 746. Restriction of the law of force for motion upon a given curve, . 747. Bonnet's theorem for combination of forces which produce a given motion (43O30), .... 748. General value of the potential for § 746 (431e), 749. Cases in which the curve of § 746 is a parabola (431,7.i9,oo_,4,3o^ii), . 750. Case in which the curve of § 746 is a conic section (432;.io), . 751. Case in which the curve of § 746 is a cycloid (432o3_oj), . 752. Case in which the curve of § 746 is a circle, or in which the surface of free motion is a sphere, . . 432-438 429 429 430 430 431 431 432 432 CHAPTER XII. MOTION OF rotation. 753. Rotation-area defined. Principle of i 754. The parallelopiped of rotation- the conservation of areas, . 433-434 1 areas, 434 ANALYTICAL TABLE OF CONTENTS. XXXVU 755. 756. 758. 759. 760. 161, 797. Rotation of a Solid Body, 434-458 The moments of inertia and the in- verse ellipsoid of inertia, . 434-436 Rotation about a principal axis pro- duces no rotation-area about the other principal axes, . . . 436 The plane of maximum rotation-area is conjugate to the axis of rotation, 436 The ellipsoid of inertia, . . . 436 Position of the axis of maximum ro- tation-area with reference to the axis of rotation in the direct and inverse ellipsoids of inertia, . 436-437 Ecler's equations for the rotation of a solid (43 7oo,3o), .... 437 The equation of living forces in the rotation of a solid (4383, 9), . 437-438 762. 763. 762-769. Rotation of a Solid Body which is srb.ject to no ex- TERNAL Action, . . . 438-443 The velocity of rotation of this solid is proportional to the correspond- ing diameter of the inverse ellip- soid (4380,,), 438 The velocity of rotation about the axis of maximum rotation-area and the distance of the tangent plane at the extremity of the axis of rota- tion are invariable. Poinsot's mode of conceiving the rotation (439i), 438-439 764. Permanency of the instantaneous axis and of the axis of maximum rota- tion-area in the body (43953), (4406), 439-440 765. Surfaces of the instantaneous axes in space (442,), . . . 440-442 766. The velocity of the instantaneous axis in the body (44 215), . .442 767. Case in which the axis of maximum rotation-area describes the circular section ; corresponding spiral path of the axis of rotation (442i9_27^), 442-443 768. Case in which the ellipsoids of iner- tia are surfaces of revolution, . 443 769. The analysis of this case may be ex- tended to that of a solid rotating about a fixed jjoint without the ac- tion of external forces, . . . 443 770-783. The Gyroscope and the Top, 443-451 770. Motion of a solid of revolution about a fixed point (444,), . . 443-444 7 71. The rotation about the axis of revo- lution is uniform, . . . .444 772. The motion of the gjTOscope (4443,), (4457_ig), .... 444-445 773. The motion of the gyroscope ex- pressed by elliptic integrals (4466,1,,), .... 445-446 774. When the velocity of rotation van- ishes, the gyroscope is a spherical pendulum, ..... 446 775. Case in which the g^'roscope de- scribes a horizontal circle, . .446 776. Major Barnard's case of the gyro- scope in which the initial velocity of the axis vanishes, . . .447 777. Case in which the azimuthal motion of the axis is reversed during the oscillation, . . . . .447 778. Case in which the axis of the gyro- scope becomes the downward ver- tical during the oscillation (44 8^), 447-448 779. Case in which the axis of the gyro- scope becomes the upward vertical during the oscillation (44819), . 448 780. Case in which the velocity of the axis vanishes for the upward vertical, . 448 781. Case in which the axis constantly approaches the upward vertical without reaching it (449j_i6), . .449 782. The theory of the top (444^), (445;), (449.^), 449-450 783. Friction in the case of the gyroscope. The sleeping of the top, . 450-451 784-791. The Devil on Two Sticks AND the Child's Hoop, . 451-456 784. Theory of the motion of the devil on two sticks (452s, 12), . . 451-452 XXXVlll ANALYTICAL TABLE OF CONTENTS. 785. The axis of the de\'il cannot be- come vertical in the general case (452,,),. . . ''. . .452 786. Case of the devil in which there is no rotation-area about the vertical axis, and in which the axis of the devil may become horizontal, 452-453 787. Case of the devil in which there is no rotation-area about the vertical axis, and in which the axis of the devil cannot become horizontal, . 453 788. Case in which the axis of the devil may become horizontal with a gyra- tion about the vertical axis ; and the corresponding case when it cannot become horizontal, . 453-454 789. Case in which the axis of the devil may become vertical, . . . 454 790. Theory of the body rolling upon a horizontal plane (45520, 2i), . 454-455 791. Peculiar motion of the hoop when it is nearly falling, . . . 455-456 792-794. Rotary Progression, Nuta- tion, AND Variation, . 456-457 792. Definition of nutation, j^rogression, and variation of axes, . . . 456 793. Accelerative forces which produce nutation, progression, or variation, 456 794. Cases of these various actions, . . 457 795-797. Rolling and Sliding Mo- tion, 457-458 795. General theory of rolling motion (457.,,), 457 796. General theory of sliding motion (4585), 457-458 797. Theory of sliding with friction ; case in which the sliding disappears, and the motion becomes that of rolling, 458 CHAPTER XIII. MOTION OF SYSTEMS. 798. Principles of power, translation and rotation applicable to all systems, . 458 799. Forces of difi'erent orders, disturb- ing forces and perturbations, . 458-459 800. Division of the system into partial systems, ..... 459 801-805. Lagrange's Method of Per- turbations, . . . 459-462 801. Method of the variation of the arbi- trary constants (460.>„), . . 459-460 802. Combination of divers modes of vari- ation (461o,9), . . . 460-461 803. Derivative of the disturbing force with reference to an arbitrary con- stant (461.i,), (462,), . . 461-462 804. Special case in which the arbitrary constants are the initial values of the variables (46 2ii,i.,), . . .462 805. Variation of the constant of power (462o..), 462 806-808. IjAplace's Method of Per- turbations, . . . 462-465 806. Direct integration of the disturbed functions which are equivalent to the undisturbed arbitrary con- stants, 462-463 807. Special case of fretjuent occurrence in planetary perturbations, . 463-464 808. Perturbations of a projectile, . . 464 809-818. Hansen's Method of Per- turbations, . . . 465-469 809. The first principle of this method, and the expression of the time as an in- variable arbitrary constant (465^), 465 810. The principle of § 809, applied to the case of §808 (465i^oo), . . .465 811. The principle of § 809, applied to the case of §807, and especially when the disturbing force has a simple, periodic form (46505), (4660, j), 465-466 ANALYTICAL TABLE OF CONTENTS. XXXIX 812. The second pi-inciple of tliis method or the application of the perturba- tions to the element of time, so that one of the functions may involve no other element of perturbation (46605), 4G6-467 813. Additional perturbation in the case of §812 of any other function (467io), . . ' . . . .467 814. Other forms of the perturbation in the first ajiproximation (46 7„), . 467 815. Case in which the function of § 812 does not involve the velocities, 467-468 816. 817. Case in which the initial values of the functions of §§ 812 and 813 are simply related to the arbitrary constants, ..... 468 818. The further development of the methods of perturbation Is reserved for celestial mechanics, . . 468-469 819-824. Small Oscillations, . 469-472 819. The theory of small oscillations Is re- duced to the Integration of a sys- tem of linear differential equations (469a;), 469 820. The superposition of small oscilla- tions, 469-470 821. Integration of the equations of § 820 (4 70ij_2,), 470 822. Admissible forms of small oscillations correspond to stable elements of equilibrium (470oj), , , 470-471 823. Independent elements of oscillation (471^,), 471-472 824. Oscillation and vibration pervade the phenomena of nature, .. ,472 825-830. A System moving ix a Re- sisting Medium, . , 4 72-475 825. Equation for the determination of the Jacobian multijiller In such a system (472^1), . . .472-473 826. The factors of the multiplier corre- spond to different laws of resistance, 473 827. Cases in which the multiplier is unity, 474 828. The multiplier when the resistance is proportional to the velocity, , 473-474 829. The multiplier when the resistance Is proportional to the square of the velocity, 4 74 830. Equation of power for a system mov- ing in a resisting medium (474i5_oi), 474 831. Motion of the centre of gravity in a resisting medium (47427,31), (4763), 474-475 832. The rotation-area In a resisting me- dium (475i,i_33), . , , .475 833. The Conclusion, 476-477 APPENDIX. Note A. On the Force of Moving I Note B. On the Theory of Ortho- Bodies, , , . . 479-480 1 graphic Projections, . . 481 List of Errata, 483-486 I Alpjiabetical Index, , 487-496 ANALYTIC MECHANICS. CHAPTER I. MOTION, FORCE, AND MATTER. § 1. Motion is an essential element of all physical phenomena ; and its introduction into the universe of matter was necessarily the preliminary act of creation. The earth must have remained forever " without form, and void ; " and eternal darkness must have been upon the face of the deep, if the Spirit of God had not first 'hnoved upon the face of the waters." 2. Motion appears to be the simplest manifestation of power, and the idea of force seems to be primitively derived from the conscious effort which is required to produce motion. Force may, then, be regarded as having a spiritual origin, and when it is imparted to the physical world, motion is its usual form of mechan- ical exhibition. 3. Matter is purely inert. It is susceptible of receiving and containing any amount of mechanical force which may be commu- nicated to it, but cannot originate new force or, in any way, trans- form the force which it has received. 1 2 — CHAPTER II. MEASURE OF MOTION AND FORCE. MEASURE OF MOTION. § 4. Uniform 3IoUon is that of a body which describes equal spaces in equal times. 5. VclocUy is the measure of motion. In the case of uniform motion it is the distance passed over in a given time, which is assumed as the unit of time, and, in any case, it is at each instant the space which the body would pass over, if it preserved the same motion durino; a miit of time. 6. If the space described by a body in the time t is denoted by s, the expression for the velocity v is, in the case of uniform motion, If the diifcrential is denoted by d and the derivative by Z>, the expression for the velocity is, in any case, dt ' 11. MEASURE OF FORCE. 7. Experiments have shown that the exertion which is re- quired to move any body, is proportional to the product of the intensity of the effort into the space through which it is exerted. This product is, then, the proper measure of the whole amount of force which is necessary to the production of the motion ; lone- established custom has, however, limited the use of the Avord force to designate the intemity of the effort, and the wliole amount of exertion may he denoted by the term iwwer. Hence, if the power P is produced by the exertion of a constant force F, acting through the space §, the expression of the force is S But if the force is variable in its action, the expression of its intensity at any point is ds 8. It is found by observation that the force of a moving body is proportional to its velocity. Thus, if m is the force of a body wdien it moves with the unit of velocity, its force, Avhen it has the velocity v, is mv. 9. Different bodies have different intensities of force when they move with the same velocity. The mass of a body is its force, when it moves with the unit of velocity ; thus, m in the preceding article, denotes the mass of the body. 10. The force communicated to a freely moving body, by a force which acts in the direction of the motion, is found to be the product of the intensity of the acting force, multiplied by the time of its action. Thus, if the mass m, acted upon by the con- stant force F, for the time t, in the direction of its motion, has its velocity increased by v, the addition to the force of the mov- ing body is mv = Ft. _ 4 — In case the acting force is not constant, the rate at which the force of the hody increases is mDt V = F. III. FOKCE OF MOVING BODIES. 11. The ijoiuer 2vith trhich a lody moves is equal to ilie iwoduct of one half of its mass multiplied If/ the square of its velocitf/. For if the hody, of which the mass is m, is acted upon by the force F, until from the state of rest it reaches the velocity V, the power P, which has been communicated to it, and which it consequently retains, must, by (Si^) *.and (43), give the equation D,P^= mDt V. The derivative of P relatively to t, is by {2^^) D,P = D,P.D,s = vD, P = mvD, v. The integral of this equation is P = hnt^, to which no constant is to be added, because the power vanishes with the velocity. (JVote A.) 12. Hence the power of a moving body is equal to one half of the product of its force multiplied by its velocity. * The form of reference here given is bj means of numbers, of which, the leading number refers to the page, and the secondary number, which is printed in smaller type, refers to the place upon the page, estimated from the top of the page, in lines of equal typographic interval. Printed marks, corresponding to these intervals, accom- pany each cojiy of the work. Thus, (814) denotes the equation which is at the 14th typographic interval from the to]> of the third page. 13. It is convenient to refer the measure of force to the unit of mass as a standard. Thus, if F is the force exerted upon each unit of mass, the force exerted upon the body of which the mass is m, is mF. With the F, used in this sense, (43) becomes D,v=^F. CHAPTER III. FUNDAME^s'TAL PRINCIPLES OF REST AND MOTION. TENDENCY TO MOTION. § 14. A system of moving hoclies may he regarded mechanically as a system of forces or iwwers, ivliich must he the exact equivalent of all the forces or iwivers which, hy simultaneous or successive communication to the hodies, are united in its formation. This results from the inertness of matter, and its incapacity to increase, diminish, or vary in any way, the power which it contains. 15. It also follows from its inertness, that matter yields instan- taneously to every force, and cannot resist any tendency to the communication or abstraction of power. With a system which is at rest, there can consequently be no tendency to the communi- cation of power. 16. The tendency of any body or system of bodies to move in any given way is easily ascertained. It is only necessary to sup- pose the system moved with the proposed motion to an infinitesnnal distance. The product of the corresponding distance, by which each body of the sj'stem advances in the direction in which each force acts, multiphed by the intensity of the force is, by § 7, the corre- sponding power which the force communicates directly to the body, and through it to the system. The zvhole amount of iwiver 2vhicJi is thus communicated hj all the forces to the si/stem, or rather Us ratio to the infinitesimal element of the proposed motion is evidentln the measure of the tendency of the system to this j^rojjosed motion. It must be observed that, when a body moves in a direction 02:)posite to that of the action of the force, the corresponding product is neQ;ative, and must be used with the ne<2;ative sim in forming; the algebraical sum, which represents the whole amount of power com- municated to the system. 17. By a skilful use of the principles of the preceding sec- tion, all the elementary tendencies to motion in a system may be determined, and, therefore, all the elements of change of motion in the system which is actually moving, or all the conditions of equi- librium in the system which is at rest. Thus, let ?«i, «?2> wzg, &c., denote the masses of a system of bodies; Fi, F[, F[\ &c, the forces which act upon each unit of ?«i; F2, F2, F2, &c., the forces which act upon each unit of ;;^2; &c, &c. ; ^1? ^!/ij f!/Tj &c., the distances by Avhich w?i advances in the direction of the forces F^, F[, Fi, &c., in consequence of any proposed motion ; ^2? ^1/25 ^/2'j &c. ; (^3, &c., the corresponding distances for the other bodies and forces of the system ; ^', the sum of all quantities of the same kind, obtained by changing the accents ; — 7 — ^i, tlie sum of all quantities of the same kind, obtained hy changing the underwritten numbers ; 2[, the sum of all quantities of the same kind, obtained by ail admissible combinations of both chano-es. The power communicated to the system by the proposed motion through nii, 1112, &c., is ^m,F,,\f, = m, {F,df, + Fid/[ + &c.) :^m^F^d/l = ;;^2 {F.^d'f.^ + F^^jl + &c.) &c. &c. ; and the whole power communicated is ^[m,F,i)f,= :E,2'nHF,d/, = :S'm^F^df^ + 2'nhF.,df.^ + &c. This is, therefore, the complete measure of the tendency in the system to the proposed motion, or of the change of motion which the moving system would experience in the direction of the pro- posed motion. But by a simple change in the values of cT/j, ^f[, cT/2, cV/2, &c., the tendency to any other proposed motion may be measured ; and, in the same way, all the elements of the change of m.otion may be definitely ascertained. II. EQUATIONS OF MOTION AND EEST. 18. If, instead of the given forces, each body were acted upon by a force in the direction of its motion, and of such an intensity as to produce the exact change of velocity which it undergoes, this new system of forces would precisely correspond to that actually imparted to the moving bodies, and would be the exact equivalent of the given system of forces. Let ^'i? ^'2? ^'39 &c. denote the velocities of the hoclies; dsi, tV6'2, dsg, &c., the distances by which, in consequence of the proposed arbitrary motion of the preceding section, the bodies advance in the actual direction of this motion ; and then from (43) DiVi, Z>«?'2? -DtVs, &c., are the intensities of the new forces relatively to the unit of mass. The whole power communicated by the new system of forces with the proposed motion becomes, then, ^im^Dtt\d^Si = miDii\dsi -f- m2DiV2(^S2 -\- &c., and it must, therefore, be equal to the expression (Tis) of the power communicated by the given forces. Hence, or by transposition When the system is at rest, this equation becomes -Zi^ii^ifT/i = o. 19. The equation (8ig) in the case of motion, or the equation (820) in the case of rest, although it appears to be a single equation, involves in fact as many equations as there are distinct elements of motion or rest in the system of bodies. For every such element gives a different set of values of cV/i, df{, df^, &c., ds^, ds^, &c., which, substituted in (Big) or (820), produce a corresponding equation. These equations, therefore, involve all the necessary conditions of motion or rest in every mechanical problem. All that remains, then, is to determine, by geometrical analysis, the various elements of motion or rest, and to integrate and interpret the algebraical — 9 — equations, into which (S^g) and (890) are finally clecomposecl. The Mecanique Anali/tiquc of the ever-living Lagrange contains the general forms of investigation with unequalled elegance and perspicuity. But the special modes of analysis, which are peculiarly adapted to the illustration and development of particular problems, have been too much neglected, and the attention of 3'outhful explorers is earnestly invited to this unbounded field of research. CHAPTER lY. ELEMENTS OF MOTION. MOTION OF TRANSLATION. § 20. A single material point may be moved to an infinitesimal distance in any direction, which may be defined by either of the methods known to geometers, by the reference, for instance, to the directions of three mutually perpendicular axes. By the known theory of projections, {Note B,) the distance by which the point advances in the direction of its actual motion, or in any other direc- tion, may be fully determined from the distances which it advances m these three directions. The three distances, moved in the direc- tions of the axes, which are simply the projections of the proposed motion upon the three axes, are the three independeiit elements of motion vjhich completely define the elemeiiiary motion of the sinr/le point. — 10 — Thus if dj) denotes the proj^osed elementary motion, if ■^5 ^7 P, denote the anf^les which this motion makes with the three mutually perpendicular axes, called the axes of x, 7/, and z, and dx, df/, dz, the projections of dp upon the axes, the expressions for these projections are, dx = cos P . dpy d>/ = cos P . dp, dsz=cos^ .dp. If, in general, g denotes the angle which the directions ofp and ^ make with each other, the distance by which the point advances, in consequence of the proposed motion, in the direction of/ is, by the theoiy of projections, d/= cos ^. dp ^ z::!-- GOS-^ .dx-r-COS-^ .d^/ -\-COS* .dz X ^ y ^ ^ z :=: J^^COS-^ .dx : ^ X ' in which 2^ denotes the sum of all the similar terms obtained by pro- ceeding from one axis to each of the others. 21. The most important of all the elementary motions of a system of bodies are those which, being independent of the peculiar constitution of the system, may be common to all systems. Such — 11 — motions must be possible, even if the bodies which compose the sys- tem, do not change their mutual positions, but are so rigidly fixed that the whole may be regarded as one solid body. It will be shown that there are but two distinct classes of such motions, namely, those of tramlaiion and those of rotation. 22. The motion of translation is that by which all the points of a body, or system of l)odies, are transported through the same dis- tance in the same direction. The projections of an elementary translation upon three rectangular axes are given by equations (IO10.12), while (IO21), is the expression of the distance by which the system, or any one of its bodies, advances in any direction, such as that of/, by reason of the proposed translation. 23. Any numl^er of different elementary translations may be supposed to be given at the same time to a system, and the result- ing motion will l3e such an elementary translation, that its projec- tion, estimated in any direction, will l^e the sum of the projections of the elementary translations estimated in the same direction. Two coexistent elementary translations may be combined geo- m.etrically by setting off from any point two lines of tlie same length with the elementary motions, and in the same direction with them ; and if a parallelogram is described upon these two lines as sides, the diagonal, which is drawn from the given point, will rep- resent in distance and direction the resulting elementary transla- tion. In the same way the geometrical resultant of the combination of three elementary translations may l:»e represented by the diago- nal of a parallelepiped described upon the lines which represent the component translations. But this parallelepiped vanishes when the three lines are in the same plane. — 12 — 11. MOTION OF ROTATION. § 24. The motion of rotation is that by which all the points of a body or system of bodies turn about a fixed line in the body, which line is called the axis of rotation. If one stands with his feet against the axes of rotation, and his body perpendicular to it, and faces in the direction of the rotation, the podtive direction of the axis of rotation is, in this treatise, regarded as lying upon his right hand, and its negative direction upon his left hand. It will be found convenient to represent a rotation geometrically by a distance pro- portional to the elementary angle of rotation, set off upon the posi- tive direction of the axis of rotation from any point taken at pleas- ure in the axis. If d^ denotes the elementary angle of rotation, and r the distance of a point of the body from the axis of rotation ; rd^ is the elementary distance through wdiich the point moves in consequence of the rotation. The form in which the subject of rotation will be here pre- sented, is not greatly modified from that which it has finally assumed in Poinsot's admirable exposition of the " Theory of the Rotation of Bodies,'' as it is printed in the additions to the Commis- sance des Temps for 1854. 25. When a hody rotates about an axis, it is, in consequence of this rotation, simidtaneoiisly rotating aboid any other axis ivhich passes through the same point, loith an angle of rotation which is represented by the pro- jection upon this neiv axis of the line ivhich represents the original angle of 7'otation. For by the angle of rotation ^ about the axis OA (fig. 1), the — 13 — point P of the axis OB, wliicli is at the distance r = P3f from the axis OA, is moved through the distance r^. Although every point of the axis OA is actually at rest, it has with respect to P, a relative motion, which is the negative of that of P. A rota- tion ^' about the axis OB gives the point N of the axis OA, which is in the plane drawn through P perpendicular to OB, and at the distance r'^PN from the axis of OB, a motion through the distance r'(3' taken nega- tively. This rotation is, then, the same with that w^iich the actual rotation produces about the axis OB, if or ^ = ^=:cosJ/Pi\^ o r = COS A OB; that is, if <3' is equal to the projection of 6 upon OB. 26. Three simidtaneous elementary rotations about three axes, tuhich pass through the same ^yoint, and are not in the same ^ilane, are equivalent to a single rotation ahoid the diagonal of a parallelojnjoed, of lohich the three lines represeiding the rotations are the sides, and the length of the diagonal represeiUs the angle of elementary rotation. For the algebraic sum of the projections of the sides of the parallelepiped upon any line perpendicular to its diagonal is zero, and, therefore, there is no rotation about any such line. Hence the diagonal is stationary, that is, it is the axis of rotation. The whole amount of rotation, being the sum of the partial rotations about the diag-onal which arise from the several rotations about the sides, is represented by the sum of the projections of the sides upon the — 14 — diagonal, which is, by the theory of projections, equal to the diago- nal itself. 27. In the same way, two simultaneous rotations about the sides of a parallelogram may be combined into a single rotation about the diagonal. In short, simultaneous cicmentari/ rotations about axes ivliich cut each other may Jje comhined in the same ivay as elemoitari/ translations. 28. To investigate the distance by which a given rotation causes any point of a body or system to advance in a given direc- tion, as that of / ; let di^ be the elementary angle of rotation about the axis of ]j and r the j^erpendicular let fall from the point upon the axis of rotation. Let a line be drawn through the given point, parallel to the projection of / upon a plane, which is perpendicular to the axis of rotation, and let i) be the perpendicular let fall upon this line from the point in which r meets the axis of rotation ; and r the angle which / makes with the direction in which the point is moved by the elementary rotation. The distance by which the point advances in the direction of /is cT/=/cos ^.8^ = /cos ^,BmP..d& in which ^ should be taken positively when the point is moved towards the positive direction of /. 29. If three rectangular axes are drawn through any point of the axis of rotation, and if — 15 — ^t?^, d^y, d^^ are the projections of d& upon these axes, the dii^- tance hy which the point {x, ?/, z) is moved in the direc- tion of the axis of x, is dx = ijd^^ — zd^y = (ycos^ — 0cos^) (5(5 = (cos ^ cos ^ — COS ^ cos ^) r()'i5 = (cos '^ cos-?^ — COS ^ COS ^^) coscc ** .r'(5"^ =:COS^/(^^ = (C0S^C0S^ COS COS ^ )?■'()' t5. \ y z z y' There are similar expressions for the distances by which the point advances in the directions of the axes of y and z, which may be found by advancing each of the letters .r, y, z^ and x to the fol- lowino; letter of the series. 30. The two last members of equation {Vo-^ divided by r'(5't5 give the following theorem ; cos ^ = cos ^ cos ^^ — cos ^ cos^, X y ^ ^ y in which the direction of ^ is that of the perpendicular to the com- mon plane of r and p, and it is taken upon that side of the plane for which, a positive rotation about it, would correspond to a motion through the acute angle from r to p. 31. If there were another system of rectangular axes, o',y\ and /, equation (ISso) applied to them would give cos := COS ^ COS COS -^ COS . X y ^ z y In this equation each of the letters x, y, z, and x might be advanced to the subsequent letter of the series, as well as each letter IG of the series x' , y', z, and x'. In this way eight other equations might be found simihir to equation (ISss). III. COMBINED MOTIONS OF ROTATION AND TRANSLATION. 32. An elementary rotation, comhined ivith an elementary translation in any direction, tvJiich is iderpendicular to the axis of rotation, is equivalent to an equal elementary rotation about an axis ivhich is parallel to the origi- nal axis of rotation. The position of the new axis is determined hy the con- dition that each of its points is carried hy the original elementary rotation as far as by the elementary translation, but in an opposite direction. For the given motions cancel each other's action upon each point of the new axis, and leave it stationary ; while the original axis advances with the elementary translation by the exact dis- tance which corresponds to the elementary rotation about the new axis. The common plane of the two axes is perpendicular to the direction of the translation. 33. Any simultaneous elementary rotations about axes parallel to each other are equivalent to a single rotation, equal to their sum, and about an axis parallel to the given axes, combined ivith an elementary translation equal to the motion ivhich any point of the nciu axis receives from their simultaneous action. This is a simple deduction from the preceding proposition. 34. Let there be three rectangular axes, such that the new axis of rotation may be that of s ; let .^1,^1, rr2,^2j &c., be the points in which the original axes cut the plane of xy ; and let (T^i, (5"^25 &c., be the elementary angles of rotation about these axes. — 17 — The elementary rotation about the axis of z is The elementary translations in the directions of the axes of x and y are by {Vl^^ d^o = — -i-'^'i^^^i- The distances through which anj^ point (.r, ?/, z) is carried for- ward in the directions of the axes, are d// =^ d'^Q -\-xd^ z=:^ — ^1 Xi di^-^-\- X -2*1 (J ^1. The points are, therefore, at rest for which = dxQ — ydi^ = ^i,yicT^i — y JE-^^d&i, = (5>o + ^■f'>^^ = — ^1 -^1 (^^1 + x2:j&^. These are, therefore, the equations of the axis of rotation, an elementary rotation about tvhich, equal to the sum of all the elementary rotations, is equivalent to the conihination of all the elementary rotations. 35. If the original elementary rotations are all equal, and if there are n axes of rotation, the equations (172) and (17ii) become (5(5 -z^ind^y, (3\r=(^i^i — 7?//) (5^^i, dy = ( — ^1 x^ -\- n .?;) d ^i . The equations (17i6) give for the single axis of rotation V y=- n ■ 36. If any of these rotations are about an axis lying in the opposite to the assumed direction, they may be regarded as nega- 3 — 18 — tive rotations about axes having the same direction as the assumed one, and may be combined algebraically in the preceding sums. 37. When the second member of equation (172) vanishes, the resulting rotation disappears, and the given elementary rotations are equivalent to the elementary translation defined by equa- tions (176). 38. Two equal rotations about axes, which are parallel, but have opposite directions, constitute a combination which Poinsot has called a couple of rotations. A couple of elementmyj rotations is, therefore, equal to an elementary translation in a direction perpendicular to the common plane of the axes, and equal to the product of the distance between the axes multiplied b?j the elemoitary angle of rotation. 39. Any simultaneous elementary motions of rotation and translation are equivalent to a single elementary rotation about an axis, combined ivith an elementary translation in the direction of the axis of rotation. For each rotation may be resolved into a translation and a rotation about an axis passing through any assumed point. But all the elementary rotations about axes passing through the same point are equivalent to a single rotation about an axis passing through the point, and all the translations are equivalent to a single transla- tion. The single translation may be resolved into two translations, of which one is parallel, and the other perpendicular to the single axis of rotation. The translation, which is perpendicular to the axis of rotation, combined with the rotation, is equivalent to a sin- gle rotation about an axis, parallel to the single axis, and, therefore, having the same direction with the remaining translation. 40. Every possible motion of a rigid system or body is equivalent to a combination of the motions of translation and rotation. This is evident, if it can be shown that, by such a combination of motions, any three points. A, B, and C, of the system, can be car- — 19 — ried to any positions, A', B', and C, in which it is possible for them to be placed. For tliree points of a rigid system not in the same straight line completely determine, by their position, that of the whole system. Now, by a translation of the system, equal to that by which A might be directly moved from A to A, the point A is actually brought to the position A. By a subsequent motion of rotation about an axis, which is perpendicular to each of the lines AB and A B', the point i? may be moved to B' ; and then by a rotation about AB' the point may be carried to C. Hence the whole motion is accomplished by one translation and two rotations. Every elementary motion of a rigid system must then be equivalent to a single rotation about an axis and a translation in the direction of the axis of rotation. This motion is perfectly rep- resented by that of the screw, whose helix causes it to advance in the direction of the axis about which it is turiiiuy:;. 41. During each instant of its motion, a rigid system rotates about an axis, which is called the mstanfaneous axis of rotation. This axis is generally varying its positio.i in the system and in space from one instant to another, which renders it difficult to form a distinct conception of the nature of the corresponding motion of the system. 42. In attempting to conceive of the motion of a rigid system, it is expedient, at first, to neglect the translation in the direction of the axis of rotation, and to assume that the motion is solely that of rotation. The successive positions of the axis of rotation in the system form by their union a surface which turns with the system ; and its successive positions in space form another fixed surface. In the motion now considered, the moving surface rolls on the fixed surface without sliding, and carries the system with it. 43. If the axis of rotation does not move perpendicularly to itself each of these surfaces is evidently a developable surface, and — 20 — in the act of rolling the line of retrogression of the one falls npon that of the other ; so that these two lines are of the same length. Upon the surfaces, developed into a plane, the two lines of retro- gression will be precisely alike. In combinins: with this rotation the translation in the direction of the axis of rotation, the surface, generated by the instantaneous axis in the moving system, remains unchanged. But the fixed sur- face, generated by the instantaneous axis, is changed ; it is still a developable surface obtained from that in which the translation is neglected, by adding to each element of the arc of the curve of retrogression, the elementary translation in the direction of the axis of rotation. In the actual motion, the moving surflice rolls upon the fixed surface, and glides simultaneously in the direction of the line of contact, so as to keep the curves of retrogression constantly in contact. In this general case, the whole length of the arc of the fixed curve of retrogression is equal to that of the moving curve aug- mented by the whole amount of translation in the direction of the axis of rotation. When the elementary translation is equal to the elementary arc of the moving curve of retrogression, but lies in the opposite direction, there is a corresponding cusp in the fixed curve of retro- gression. A point of inflection in the curves of retrogression generally cor- responds to a change in the direction of the rotation. A similar combination of the translation with the rotation can be introduced into the general case of motion. 44. When either of the surfaces of the instantaneous axis is a cone, the curve of retrogression is reduced to a point which is the vertex of the cone. When both of the surfaces are cones, there is no translation in the direction of the axis. — 21 — When either of the surfaces is a cjdinder, both surfaces must be cylinders; and the lines of retrogression, removing to an infinite distance, cannot be used for guiding the motion of translation. But in this case, a section may be made of one of the cylinders per- pendicular to its axis, and in the actual motion the moving cylinder will move so as to keep the point, in which the perimeter of this section touches the other cylinder, upon a curve proj)erly drawn upon that cylinder. 45. The general motion of a rigid system may be conceived as a translation, equal to that of any one of its points assumed at will, combined with a rotation about an instantaneous axis of rotation passing through the point. If the translation is neglected, the rota- tion is effected as in § 42 by rolling a cone, of which the assumed point is the vertex, and which carries the system with it, in its motion, about a fixed cone, of which the same point is the vertex. The translation may be simultaneously effected by moving the two cones in space, with a translation equal to that which belongs to their vertex in the actual motion of the system. 46. For all the points of the instantaneous axis in each of its positions, the corresponding centres of greatest curvature of either of the conical surfaces which it describes, are all upon the same straight line passing through the vertex. In the case of the right cone, or of the right cylinder, the axis of revolution is the line of the centres of greatest curvature. In all these investigations the plane may be regarded either as a cylinder of infinite radius, or as a cone, of which the angle at the vertex is equal to two right angles. 47. The elementary rotation of the system may be conceived as deconiposed into two elementary rotations about the lines of the centres of greatest curvature as axes of rotation. By the rotation about the line, which unites the centres of the fixed surface, the — 22 — instantaneous axis receives its elementary motion in space, and is carried to its proper position upon tlie fixed surface. By the rota- tion about the Une wliich unites the centres of the moving surface, the system receives that additional rotation which is required to turn the moving surface into that position in which it may have the proper line of contact with the fixed surface. Each of these rota- tions produces a sliding of the moving upon the fixed surface ; but as the sliding produced by the one is just equal and opposite to that ]3roduced by the other rotation, the two rotations cancel each other's action in this respect, and there is no sliding in the combined motion, but a simple rolling of one surface uj)on the other. 48. Let «y be the acute angle which the instantaneous axis of rota- tion makes with the line of the centres of curvature of the fixed surface ; «„, that which it makes with the line of the centres of cur- vature of the moving- surface, this angle heing positive when the two lines of the centres are on opposite sides of the instantaneous axis, and negative, when they are upon the same side ; dm the elementary angle by which the instantaneous axis changes its direction ; (5" ^f the elementary angle of rotation about the line of cen- tres of the fixed surface ; and d (5,„ the elementary angle of rotation about the line of cen- tres of the moving surface. Since the instantaneous axis must be carried forward by the rotation about the fixed axis, and backward by the rotation about 23 the moving axis just as far as its actual change of position, its ele- mentary angle of change of direction is d(i) =id &j: sin «/= d ^,„ . sin a„^ . But the combination of the two rotations about these axes gives the actual rotation about the instantaneous axis, and there- fore, d" ^ = (5" <5y. cos c(f-\-^ ^m • COS a^ =^ (cot «/-|- cot «„,) d ci» sin («/+«,„) V Sin Ujr sin «,„ 49. When the surfaces described by the instantaneous axis are cylinders, let ()j- and ()„j be the respective radii of greatest curvature of the fixed and moving surfaces at any point of their mutual contact ; and djy the elementary distance which the instantaneous axis moves in a direction perpendicular to itself The conditions of the motion of the instantaneous axis give the equations dp = Qyd <5^= ± Q,n ^ ^\n ; in wdiich the upper sign corresponds to the case where the lines of the centres of curvature are upon opposite sides of the instanta- neous axis, and the lower sign to that in which they are upon the same side. The rotation about the instantaneous axis is d6 = dd^+di\. — 24 — lY. SPECIAL ELEMENTS OF MOTION AND EQUATIONS OF CONDITION. 50. The variation of each independent element of position of a system gives an independent element of motion. But the ele- ments of position are various, and must be selected in each case with special reference to the problem under discussion. It often occurs that parts of the system are rigidly connected ; such parts are themselves rigid systems, and subject only to motions of trans- lation and rotation, and, therefore, none but such elements are required for the investigation of their motions. Points of the system are sometimes restrained to move upon given surfaces, and, in this case, it may be expedient to introduce elements of position dependent- upon the principal lines of curva- ture of these surfaces, or elements, in reference to which the sur- faces are peculiarly simple or symmetrical. Points of the system may be compelled to preserve simple geometrical relations to each other, which may suggest appropriate elements of position to the skilful analyst ; or he may find indications to direct his choice in the very nature of the motion itself 51. It is often desirable to adopt a combination of elements of position which are not wholly independent of each other, but are subject to certain mutual restrictions. These restrictions, when they are expressed algebraically, are called equations of condition. They may assume the differential form of ecjuations between the elementary motions ; or they may be finite equations between the elements of position, in which case they may be reduced by differ- entiation to equations between the elementary motions. By means of the equations of condition, as many of the ele- ments of motion may be determined in terms of the rest as there — 25 — are equations of condition ; and the remaining elementary motions may be regarded as independent of each other. 52. Instead of introducing into the equations (Sig) and (820) of motion and rest the special values of ds^, ds2, &c., cT/*i, ^f^, kc, for each particular element of motion, their general values may be found in terms of all these elements. When the elementary motions are wholly independent, their coefficients in these equa- tions give, when they are equalled to zero, the same equations which would have been obtained by the special investigations. But when the elements are not independent, all, except the inde- pendent elements can be eliminated by means of the values given by the equations of condition. The equations (Sig) and (820) of motion and rest, on account of their differential form, are necessarily linear in reference to the ele- mentary motions ; and the differential equations of condition are likewise linear. The proposed elimination may therefore be con- ducted by the mdJiod of multipliers. By this process each differential equation, multiplied by an unknown quantity, is to be added to the given equation of motion or rest. The unknown multipliers are to be determined by the conditions that the coefficients of the elemen- tary motions, which are to be eliminated, become equal to zero. Since the remaining elementary motions are indejDcndent of each other, their coefficients must also be equalled to zero. In the sum, therefore, obtained by the addition of the equations, each of the coefficients of the elementary motions is equal to zero. The num- ber of unknown quantities is increased in this process by that of the unknown multipliers ; but, because there are as many equations of condition as there are multipliers, the whole number of equations, including the equations of condition, in their finite form, is just sufficient to determine the values of the multipliers and of all the elements of position. 4 — 26 — 53. Let l3e one of the equations of condition in its finite form ; and let its differential form 138 dL^ = 0. Let also, I be tlie unknown multiplier by which it is to be multiplied. The sum obtained by adding the similar products of all the equa- tions of condition to equation (S^g) or (820) is which is the equation of motion or rest, and in which the general values of ds^, (T/j, &c., are to be substituted, and the coefficient of each elementary motion is to be equalled to zero. 54. Each equation of condition becomes the equation of a surface, to which any one of the points w^hose elements of position occur in the equation is restricted, provided that, for the moment, the variations of all the other elements are neglected. Since the point is restricted to move upon the surfiice, it cannot move in the direction of the normal to the surface. Let a system of three rec- tangular axes be adopted, and let i\^be the normal to the surface. Its variation, arising from the variation of coordinates, which may be regarded as the elements of position of the point, is If the equation of the surface is (263), Avith the omission of the num- — 27 — bers written below, which may be neglected in the general discus- sion, its variation is Let, then, and the angle, made by the normal with one of the axes, is given by the equation X _ D,L COSy -JJ-) which substituted in (2629) gives dX= M ~ M ' Hence the equation of condition with its multiplier ma}- be writ- ten in the form WL = ).3IdX=0; and this form may be substituted in the equations (2612) and (2G13) of motion and rest. — 28 CHAPTER V. FORCES OF NATURE. I THE POTENTIAL AND ITS RELATIONS TO LEVEL SURFACES, THE POSITIONS OF EQUILIBRIUM, AND THE POSSIBILITY OF PERPETUAL MOTION. § 55. It appears, at first sight, to be inconsistent with the assumed spiritual origin of force, that the principal forces of nature reside in centres of action, which are not thinking beings, but parti- cles of matter. The capacity of matter to receive force from mind in the form of motion, contain and exhibit it as motion, and commu- nicate it to other matter, under fixed laws, is not, however, less diffi- cult or more conceivable than the capacity to receive and contain it in a more refined and latent form, from which it may become mani- fest under equally fixed laws. It is only, indeed, when force is thus separated from mind, and placed beyond the control of will, that it can be subject to precise laws, and admit of certain and reliable computation. 56. The laws of the development of power in nature are of two classes. In the one class, the forces depend solely upon the relative positions of the bodies, and may be called fixed. In the other class, the forces depend, not only upon the positions of the bodies, but also upon their actual state of power, especially upon the velocities and directions of their motions \ and these forces may be called vtiricMc. 57. The most fruitful and enlarged view of the fixed forces of — 29 — nature, and one which peculiarly corresponds to their laws of action so fiir as they have been observed, is to regard them as the mani- festations of the di/namic situation of the bodies which exhibit them. The dynamic situation depends solely upon the masses and posi- tions of the bodies; it is a condition oi^ form, and its research is a problem of pure geometry. The algebraic function which embodies the idea of the dynamic state is called the potential. Its complete investigation and determination involves the solution of all the problems which can arise in regard to the power and the conditions of force of all systems, whether they are at rest or in motion, so far at least as the fixed forces of nature are concerned. The amount of power of a system is not to be inferred from its situation, although there is a certain measure of power appropriate to that situation. It is this latter power which is expressed by the potential of the system, and expressed as a function of all the ele- ments of position, by which the situation is defined. 58. TJic power of a moving system increases or decreases vitJi the poiver tvMch t>elongs to its situation, and tlie increase or decrease of its power is measured hj that of its potenticd. 59. Hence, if a sj^stem moves from a state of rest, its power is constantly equal to the excess of its potential over the initial value of the potential ; and it can never arrive at a position in which the potential would be less than its initial value. No system, indeed, can move to a situation in which the potential would be diminished more than the initial power of the system. 60. When a system is in a permanent state of rest which the actual forces do not tend to disturb, its dynamic condition is such, that the power of the system is not changed by a slight change of position. Hence, The potenticd of a system tvhich is in equilibrium, is genercdly a maxi- mum or a minimum. The exceptional case of a condition of indiffer- — 30 — ence rarely occurs in nature ; but even this case may be philosojDhi- cally regarded as the combination of a maximum and minimum, or as the result of several such combinations. 61. When a moving system passes through a position of equi- librium, or a position which is one of equilibrium in reference to the element of position Avith which the system is changing its place, the jDower of the system is either a maximum or a minimum, or in a condition of indifference. 62. When a system, in a state of rest, is placed very near the position of equilibrium, it cannot tend to move away from the posi- tion of equilibrium, if the potential of that situation is a maximmn relatively to the element by which the system is removed from it ; and it cannot tend to move towards the situation of equili- brium, if the potential is a minimum for the same element. On this account the equilibrium is stable, in reference to those elements for which the potential is a maximum, and it is iinstahle in reference to these elements, for which the j)otential is a minimum. 63. As when a function changes in consequence of the change of any one of its variables, the maxima and minima succeed each other alternately ; in the motion of a system, the positions of stable and unstable equilibrium, relatively to the element of change of position, succeed each other alternately. Situations of equilibrium of indifference may be interposed without disturbing the order of succession of the situations of stable and unstable equilibrium. If the system returns to its initial position, it must have passed through an even number of such situations of equilibrium, rela- tively to the element of change of position, half of which must have been positions of stable, and the other half positions of unstable equilibrium. In general, these situations will not be positions of absolute equilibrium, but only such in reference to the changing element of motion. — 31 — 64. Fixed forces might easily be imagined different from those of nature, and in the action of which the power of a moving system would depend upon its previous situations as well as upon its actual position. "With such forces the increase or decrease of power of a s^'stem would vary with the path which it pursued in moviuo; from one situation to another, and would be greater bv one path than by another. The change of power for each element of any given path, would still be computed by the process of § IT, and thence the whole change of power would be obtained by inte- gration. If the motion of the system were reversed, and it were carried back through the same path to its initial position, its initial power would be restored. If, of two courses, by which a system could move from one situation to another, it were forced to go by that through which it would arrive, with the greater power at its final position, and if it were then made to return to its initial posi- tion by the other path, it would return with an increased power ; if it were afj;ain to move throuu'h the same circuit, it would a<2:ain return with an equal additional increase of power ; and, by succes- sive repetitions of this process, the power might be increased to any, even to an infinite amount. Such a series of motions would receive the technical name of a perpetual motion, by which is to be under- stood, that of a system which would constantly return to the same position, with an increase of power, unless a portion of the power were drawn ofi" in some way, and appropriated, if it were desired, to some species of work. A constitution of the fixed forces, such as that here supposed, and in which a perpetual motion would be pos- sible, may not, perhaps, be incompatible with the unbounded power of the Creator ; but, if it had been introduced into nature, it would have proved destructive to human belief, in the spiritual origin of force, and the necessity of a First Cause superior to matter, and would have subjected the grand plans of Divine benevolence to the will and caprice of man. net oJ — 65. A surface, for each of whose pomts the potential has the same value, may be called a level surface. A level surface may be drawn through any point in space. Since the potential of every finite system of nature vanishes for an infinitely distant point, all the level surfaces of nature are finite, and, returning into themselves, include a space tvhich they tvholli/ surround, ivitli the exception of those level surfaces for tvhich the potential is zero. 66. A material point, placed upon a level sm^face, has no ten- dency to move in the direction of the surface, because there is no increase of power in such direction. The tendencij of a material point to motion is, therefore, 'perpendicular to the level surface upon ivhich it is placed. 67. If two level surfaces are drawn infinitely near to each other, a material point, flaced upon cither of them, tends to move in the direction, from the surface of the less potential towards the other, tuith a force ivhich is measured hj the quotient of the difference of the potentials of the two surfaces, divided hj their distance apart. Hence, if the surfaces are, throughout, at the same distance apart, the disposition to motion is everywhere the same. If the surfaces w^ere to intersect each other, the tendency to motion in the line of intersection would be infinite ; but, since there is no such infinite tendency to motion in nature, each level surface of nature must he whollij included within every other level surface, ivithin ivhich any portion of it is included. For the same reason, the potential in nature is always a continuous function. 68. Within each level surface of nature there must be a point or points of maximum or minimum potential. A continuous curved line, drawn perpendicularly to each of the level surfaces which it intersects, represents a line of action or tendency to motion, and every such trajectory must finally terminate in one of the included points of maximum or minimum potential. Each of — 33 — these points may then be regarded as a centre of action, towards, or from which, all motion tends along the various trajectories, accord- ing as the point is that of a maximum or a minimum potential. 69. If the 'potential has a constant value for any portion of space, this same constant value must extend throughout all that space, including this portion, for ivhich the potential and all its derivatives are finite and con- tinuous functions. For, in order that the potential may be absolutely constant for any finite extent, however small, all its derivatives must vanish. But it follows, from Taylor's Theorem, that the difference of the value of the potential for any portion of space, for wdiich it is continuous and finite, as well as all its derivatives, is a linear function of its derivatives at any point of that space. The difference of the potential, therefore, vanishes, when all the deriva- tives vanish and the potential is constant. The portion of space, for which the derivatives are originally assumed to be constant, must be a solid, having the three dimen- sions of extension, in order that this theorem be applicable. 70. Throughout any such portion of space, in which the potential is constant, there can be no tendency to motion in any direction. In such extent, therefore, there can be no mass of matter, for it is contrary to experience that there should be matter where there are no dynamical phenomena. 71. In all the observed laws of material action, the potential, which belongs to the action of each particle of matter, is finite and continuous, as well as all its derivatives, for the whole extent of space exterior to the particle. Hence, the potential and its derivatives, for every system of nature, are finite and continuous functions throughout any portion of space which contains no material mass. 72. Hence, it follows, that for every finite system of nature, any portion of space, in ivhich the potential is constant, must he finite, and hounded on all sides hy material masses. This portion of space cannot — 34 — extend to infinity, because, if it were to have such an extent, the finite mass, which w^oukl be its inner limit, would exhibit no external indication of force ; whereas, it is obvious that no matter can ever have been observed, except by such a manifestation of its existence. 73. There are forces in nature which are temporarily fixed, and for which the j)otential may vanish throughout all space exterior to the limit in which the centres of action are contained. 74. The difference between the values of the potential for any two points may be computed by supposing a unit of mass to move from one point to the other upon any line taken at pleasure, and determining the change of power which it receives from this motion. The change of the potential may be computed for each force separately, and, in making the partial computations, it is sufficient to suppose the unit of mass to move from the level surface of one point to that of the other, and one of the perpen- dicular trajectories may be taken for the path of this motion. 75. If, in any system, F, F', &c., are the forces ; /,/', &c., the directions in which they act ; and i2 is the value of the potential ; the general expression of the potential for any point of the system is a^^'/Fdf, in which the limits of integration extend from the values of/,/', &c., w^hich correspond to the position of the point, to infinity. The expression for the tendency to motion in any direction, as that of ;?, is D,ll = D,2'fFdf. II. COMPOSITION AND RESOLUTION OF FORCES. 70. No phenomenon is observed, in which a single force acts freely by itself. In all cases, various forces are combined ; and it is important, therefore, to ascertain what are the dynamical results of such combinations. 77. A single force acts, at each point, perpendicularly to its level surface, with an intensity which is measured by the derivative of the potential, taken with reference to the element of direction of the force. The intensity of its action, in any other direction, is measured by the derivative, with reference to the element of that direction. If another level surface is drawn infinitely near the one which passes through the point, the action in any direction is inversely proportional to the length, intercepted by the surfaces, upon a straight line drawn in the given direction. But the surfaces may, for this purpose, be considered as reduced to their parallel tangent planes at the given point • and the length, intercepted between two parallel planes, upon a straight line, is proportional to the secant of the angle which the line makes with the perpen- dicular to the plane. Hence, the action of a force in the direction of any line, is proportional to the cosine of the angle which it makes with the direction of the force. If, then, upon a straight line drawn in the direction of a force, a length is taken to represent the intensity of the force, the action in any direction is represented by the projection of this length upon that direction, or by using the word force for the representa- tive of the force, the proposition becomes, that the action of a force in any direction is the projection of the force upon that direction. 78. When several forces act upon a point, their total action in — 3G — any direction is the algebraic sum of their projections upon that direction. 79. When three forces, ivhich are not in the same plane, act iq^on a point, their combined action is equivalent to that of a single force, which is represerded in magnitude and direction ly the diagonal of the parallelopiped constructed upon the three forces. For the algebraic sum of the projections of the forces upon any direction perpendicular to the diagonal, is zero, while that of the projections upon the diagonal is the diagonal itself 80. All the forces ivhich act upon a point, are equivalent to a single force, which is called their resultant. For a single point can only tend to move, with a certain intensity, in some one direction, however various may be the forces which act upon it ; and any such tendency to motion can be produced by one force acting upon the point. The actions of all the forces in three directions which are perpendicular to each other, can be found by § 78 ; and these three partial forces can then be combined by § 79 into one force which will be the resultant. But the following method of finding the resultant illustrates the use which may be made of the level surfaces. 81. In considering the action of a force upon a fixed point in space, the variable character of the force for other points of space may be neglected, and its level surfaces may be regarded as parallel planes perpendicular to the direction of the force. Thus, it may be assumed that Ff is the potential of the force F, which acts in the direction : of/; for Df{Ff) = F, is the intensity of the force ; and Ff=z a constant, or / = a constant, — o< — is the equation of a plane perpendicular to /. Hence, the potential of all the forces which act upon the point, is If then Pq is the resulting force resolved in the direction of q ; if p is the direction of the resultant, and P is the resultant ; the value of either of these forces is represented by the formula P^ = D^n = ^'FDJ= ^'Fcosf. But, by putting the condition that ]? is perpendicular to the level surface, for which the potential is constant, gives X L =^ p. L' Hence the value of the resultant is P = i>,i2=.^.. D. .^D, ,x ==-^,i>,i2cos \P ■ X — ^ PI L ~ L ~ L T J ( ^ p5 !\ 82. By an elementary motion of translation, each point of a system is carried to the same distance in the same direction ; the potential of the system is changed, therefore, precisely as if all its points were united in one, and all the forces applied at this point. The tendency of a system to any motion of translation, is, then, the same as — 38 — that ivhich imild arise from the action of a single force, equal to the resultant of all the forces, supposed to he applied at the same point. 83. The moment of a force, ivith reference to a point, is the product of the force multiplied by its distance from the point. The moment of a force, with reference to a line, is the product of the projection of the force upon a plane perpendicular to the line multiplied by the distance of the force from the line. The moment of a force, with reference to a line, may be rej)resented geometrically by a corresponding length taken upon the line, and the name of the moment may be given to its geomet- rical representative. The moment of a force, ^Yith reference to a point, is the same with the moment, with reference to the line, which is drawn through the point perpendicular to the common plane of the point and the force. 84. The moment of a force, ivith reference to a line passing through a point, is equal to the projection upon the line of the moment, tvith reference to the point. For the moment, with reference to the point, is equal to double the area of the triangle, of w^hich the base is the force, and the altitude is the distance of the force from the point ; and the moment, with reference to the line, is equal to double the area of the triangle, of which the base is the projection of the force upon the plane perpendicular to the line, and the altitude is the distance of this projection from the line. But the latter of these triangles is the projection of the former upon the plane, and its area is equal to the product of the area of the former triangle, multiplied by the cosine of the angle of the planes of the two triangles. But the lines upon Avhich the moments are represented, being resj)ectively perpendicular to these planes, have the same mutual inclination. The moment, with reference to the line, is, therefore, equal to the product of the moment, with reference to the point, multiplied by — 39 — the cosine of the mutual angle of the moments ; that is, it is equal to the iDrojection upon the line of the moment, with reference to the point. 85. Hence it follows that the moments of forces, with refer- ence to points, may be combined by the same processes in Avhich the forces themselves are combined, and that all the moments, ivith reference to a jwint, may he combined into one resultant moment. 86. The tendency of the force F, of which the potential is Ff, to produce an elementary rotation, di), about a line p, is But if (1426) gives Q is the distance of F from ^7, A/=(>sin^; the projection of jP upon the plane perpendicular to^j, being i^sin-^, the tendency to rotation about/? becomes Q Fmi^l. = the moment of F w^ith reference to j) ; that is, the moment of a force, tvith reference to a line, is the measure of its tendency to ^produce rotation ahout that line. 87. The direction of the positive moment must be assumed to be the same with that of the axis, about which the tendency to rotation of the force is positive. 88. The resultant moment of all the forces of a system, loitli reference to a point, is the measure of their tendency to produce rotation ahout that point. Hence, the one force, of which the moment is equal to the resultant moment, has the same tendency to produce rotation. . — 40 — 89. The resultant moment of all the forces which act upon a point, with reference to any line or to any other point, is the same with the moment of their resultant. For the point upon which the forces act tends to move in the direction of their resultant, with a force equal to its intensity, and its moment is, therefore, the measure of the tendency to motion. 90. The moment of a force, with reference to a line y, is equal to its moment, with reference to a parallel line />, increased by the moment of an equal and parallel force, acting at any point of the line p. For the distance of the original force from the line y, is equal to its distance from the line p, increased by the distance from^' of the parallel force passing through jk>. 91. Hence the resultant moment of any forces, with reference to a line p, is equal to their resultant moment, ivith reference to a parallel line p, increased hy the moment, with reference to p, of equal and parallel forces acting at any point of the line p. 92. The residtant moment of any forces, ivith reference to a point 0' , is equal to their residtant, ivith reference to a ^^oint 0, increased by the moment, ivith reference to 0' , of equal and parallel forces acting at 0. For this proposition is true for each pair of the parallel axes of two parallel systems of three rectangular axes, of which the points and 0' are the respective origins. 93. A coujyle of forces is a system of two parallel and equal forces which act in different lines. 94. The moment of a couple of forces has, for every point of space, the same value, ivhich is equal to the moment of one of them for any point of the other. For two forces, equal and parallel to them, applied at any point, destroy each other's action, and their resultant vanishes. 95. The tendency of a couple of forces to produce rotation about a point, is the same as that of any system of forces, when its moment is equal to the resultant moment of the system, with — 41 — reference to the point. But the couple has no tendency to produce a transhxtion ; whereas the resultant of a system of equal and parallel forces, acting at the point, has all the tendency of the system to produce translation, but none to produce rotation about the point. Hence, the three forces, of zuhich one is the resultant of the equal and parallel forces acting at a point, and the other two constitute a couple, of 2uhich the moment is the same zuith the resultant moment, ivith reference to the point, f idly represent any system of forces in their tendency to produce rotation and translation. 96. Since the position of the couple of forces is quite arbi- trary, one of the pair may be taken to act at the same point with the resultant of all the forces; and, by combining it with the resultant, the system of three forces may be reduced to two. 97. A point can always be found in space, for which the moment of a given force has any assumed magnitude, and any direction which is perpendicular to the force. Because the distance of the point from the force, which is one of the factors of the moment, may vary from zero to infinity, and its direction from the force may be that of any perpendicular to the force. Hence, if the resultant moment, with reference to a point 0, of any system of forces, is decomposed into two moments, of which one has the same direction with the force, and the other is per- pendicular to it, another point 0' can be found, for which the moment of the resultant, acting at 0, is, in amount and direction, the negative of that component of the resultant moment for 0, which is perpendicular to the resultant. For the point 0' , there- fore, the resultant moment, coincides in direction with the result- ant itself; and of the three corresponding forces which represent the tendency of the system to produce rotation and translation, the plane of the couple is perpendicular to the direction of the result- ant. 6 — 42 — 98. If all the forces lie in the same plane, for any point of the plane the moment of each of the forces is perpendicular to the plane, and, therefore, the resultant moment is perpendicular to the plane. But the resultant of the parallel and equal forces acting at the point must, if it does not vanish, lie in the same plane, and be perpendicular to the resultant moment. If, then, the resultant does not vanish, a point of the plane can be found for which the result- ant moment vanishes. 99. If all the forces are parallel, the moment of each of them, for any point, lies in the plane which is drawn through the point perpendicular to the forces. But the resultant of the parallel and equal forces, acting at the point, has the same common direction with them, and is, therefore, perpendicular to the resultant moment. If, then, the resultant does not vanish, a point can be found for w^hich the resultant moment vanishes. Hence, if all the forces of a sf/stem lie in ilie same plane, or if they are all parallel to each other, their tendency to produce translation or rotation is eqidvalent, either to that of a single force, or to that of a couple of forces. 100. If of any system of forces, and for a point J/ is the resultant moment, R the resultant of equal and parallel forces acting at 0, Mp and R^ the projections of J/ and R upon the direction of 77, and if the same letters accented denote the same quantities for the point 0', and if X, y, and z are the rectangular coordinates of 0' with reference to 0, the value of the moment of the forces for either of the axes passing through 0' is, M'^^M^ — zR^-^^yR^. — 43 — But if the direction of the axis of z is assumed to be the same with that of R, these moments become 3C=3I,. The coordinates of the points, for which the resultant moment has the same direction with tlie resultant, are 71/, My^ ^"~ 'R'^— R' 101. The number of forces which is required to produce any of the special effects of a given system of forces, is usually much less than the whole number of those which actually concur in their production. The mode of analj'sis, by which the requisite forces may be ascertained, is, in most cases, quite as simple as that by which the effects of rotation and translation have been investi- gated. III. GRAVITATION, AND THE FORCE OF STATICAL ELECTRICITY. 102. Gravitation is, among all the forces of nature, conspicuous for its universality, and the grandeur of the scale upon which it is exhibited. Each particle of matter is an elementary centre of action for the force of gravitation, and all the level surfaces for each imrticU are spherical surfaces, of which the particle is the centre. The value of the potential for any particle, is inversely proportional to the distance from the particle, and for different particles it is proportional to the mass of the particle. 103. Another force which seems to be equally universal with gravitation, and of which gravitation has been, perhaps justly, — 44 — regarded as a residual force, and which is subject to the same law, in respect to distance from each elementary centre of action, is that of statical electricity. This force, however, is endowed with duality, and consists of tivo forces, of ivhicli one has a positive, and the other a negative potential. Both forces are usually combined with equal intensity, in the same centre of action, so as to neutralize each other's influence, and thus lie dormant. With each of these the poten- tial is positive in reference to electricity of the other Jcind, and negative iiith reference to that of the same Jcind. The tendency to motion, arising from one kind of electricity, is exactly equal and opposite, then, to that which arises from the action of an equal intensity of the other kind, distributed in the same way. 104. The action of electricity upon the mass of a particle is indirect ; the direct action is upon the electricity associated with the mass. In most bodies the electricity yields with more or less facility to this action, leaves the particle with which it is originally combined for another particle, and finally assumes such a form of distribution within and upon the hodg, tJiat the tendency to motion shall nowhere exceed the resistance to motion. Bodies in which there is no resistance to the motion of electricity are called perfect conductors; while those in which the resistance is infinite are called perfect non- conductors. 105. Let dm denote the mass of a particle of matter in the case of gravitation, or the value of its potential at the unit of distance, in the case either of gravitation or elec- tricity ; da, the element of volume of the mass ; Jc, the density of the matter, in the case of gravitation, or the intensity of the force of electricity, compared with the unit of intensity ; — 45 — /, the distance from the particle ; cUl, the value of the potential for the particle ; the expression of the potential for the particle is , ^ dm hda The general value of the potential for the whole body is 106. With reference to a system of three rectangular axes, let X, I/, z, be the coordinates of the point in space, for which the potential is 12, and ^, 7j, i', those of the particle. Adopt also the functional notation The derivatives of/ and /~^ are A/=COSy = -— r-. j^,f^ P-(—^^ _^ Hence r f — l-f 3cos%^ — J. . 1 _3_l_32;cos2- — 46 — and, therefore, This last equation, which is called Laplace's equation, only applies to that extent of space for which the derivatives of the jDotential are continuous functions, that is, where there are no centres of action ; but, where there are centres of action, it requires a modification which will soon be investigated. The integration of this equation, combined with peculiar considerations in special cases, gives the value of the potential for all the problems of gravitation or statical electricity. 107. T/ie tendency to motion, resulting from the gravitating or electrical action of a particle of matter, heing normal to the level surface, is directed in the straight line draimi to the imrticle. Its intensity is the derivative of the potential, and expressed by the equation. The force of the gravitating or electrical action of a particle of matter, is, therefore, inversely froportional to the srpiare of the distance from the particle. It is attraction in the case of gravitation, or hetween electricities of opposite lands, and repidsion hctiuccn electricities of the same Jdnd. ATTRACTION OF AN INFINITE LAMINA. 108. The investigation of the potential of a lamina of uniform density, and included between two infinitely extended planes, is simplified by the consideration, that it must have the same value for all points of space which are at the same distance from either surfiice of the lamina. Because all such points are similarly situ- ated with reference to the lamina, on account of its infinite extent. Hence, if either surface of the lamina is adopted for the plane of g z, — 47 — the derivatives of the potential, with reference cither to y or z must vanish, and Laplace's equation becomes The integral of this equation gives the value of the potential, for a point external to the lamina, or upon its surface, in which A and B are arbitrary constants. 109. The level surfaces are the planes determined by the equation (477), when il is the constant value of the potential for the level surface. 110. The action of the lamina upon any external point, is in a direction perpendicular to either surface, and its force of attraction or repuhion is constant upon all points, for it is given by the equation I),S2=A. 111. The values of A and B in any special case must be ascertained by direct integration. The integration indicated in (458), gives an infinite value of the potential, whereas the integra- tion of its derivative, with reference to x, gives A itself, in a finite form, which shows that the infinite portion of the potential belongs to B. The integration for finding the derivative of the potential is effected by putting Q=ffimJ, = the projection of/ upon the plane of ?/:s. a = the thickness of the lamina ; whence f=^{x — ^)secj, Q = [x — ^')tan'^. — 48 — da = Qclod^d^, ^■cosj 7^ =-J-- i J?"^"^- = —2jia/c = A. This value of A corresponds to a positive value of .r, but for a nega- tive value of X its sio:n must be reversed. 112. For a point situated within the lamina, a plane may be drawn through it parallel to the superficial planes, and dividing the lamina into two partial laminae, of which the thicknesses are a: and a — X. Hence, the value of the derivative of the potential is DJ2 = — 27i/cx-{~2Tt/c{a — x) =.2nk{a — 2x). poisson's modification of Laplace's equation for an interior point. 113. The modification which is required of Laplace's equa- tion, in order that it may be apj)licable to any point of an acting mass, must be the same for all cases. For it would not be needed, if the point of action were contained within any extent, however small, of void space. It depends, therefore, exclusively upon the infinitesimal portion of matter at the point, and is unaffected by any variations in the form and extent of the acting body. It need be investigated, then, in only a single case. Now the derivative of (48i6) gives which substituted in Laplace's equation gives for an internal point — 49 — of the infinite lamina, wliicli is, therefore, the required modification of this equation. This modified equation, in which Jcq, denotes tlie value of h at the point of action, is applicable, as remarked by Sturm, even when the point is exterior to the body. This same geometer has observed that, by supposing the value of Jc gradually to shade off from its value within the body to zero, this graduation occurring within an infinitely small extent, so as not sensibly to interfere w^th the actual phenomena of nature, the potential and it^ differential coeffi- cients may become continuous functions. It must be further observed, however, tliat this imaginarj^ graduation must extend throughout all space, although Jc must have an infinitesimal value where there is no portion of active force ; for if it were to vanish throughout any finite portion of space, however small, the reason- ing of § 69, would prove that all the derivatives of the potential were not finite and continuous. ATTRACTION OF AN INFINITE CYLINDEE. 114. The investigation of the potential of an infinite cylinder is simplified by the consideration that its value must be the same for all points situated upon the same straight line parallel to one of the sides of the cylinder. If this direction is adopted for the axis of s, the derivative of the potential, with reference to .e, must vanish, and Laplace's equation becomes The integral of this equation is 7 — 50 — in which cF and S^j are arbitrary functions, and must be determined for each case by special considerations. 115. The level surfaces are the cylindrical surfaces, of which (49go) is the general equation, if i2 has the constant value belonging to that surface. 116. The attraction in the direction of the axis of x is in which the accents denote the derivatives of the functions, with reference to their explicit variables. The attraction^n the direction of the axis of^ is D^ii = [W'i^ + i/vC^) - g!((^ -yv'~.)]vC"i. The whole action is, then, V/[(A)^ + (i>,)^]-<2 = 2v/[9^'(^ +^V^).3^;(^-y\C^)]. 117. When the jwint of action is so far from the cijllnder that the square of the linear dimensions of the hase can le neglected, in comparison ivith the square of the least distance of the point from the cylinder, the problem can be greatly simplified. Find in this case a line parallel to the axis of z, of which the co'drdinates a and h, with reference to the axes of x and y, are determined by the equations am. O in O m J{V — Z*) = = / 1] — Im. in U m This line may be called the axis of gravity of the cylinder, and its position is wholly independent of the directions of the axes of 90 and y. For the conditions by which this axis is determined will — 51 — give, with regard to any other axis of x, with reference to which the notation is distinguished by the subjacent numbers, r(i'i— «i)=r(^— «)cos^ +r (j/— ^)cos^ =o. If the axis of gravity is, then, assumed for the axis of s, the equations (5025_26) become or 118. Since, from the nature of the cylinder, the functions which are here to be integrated are independent of t, these equations give 119. Let the perpendicular from the point of action upon the axis of 2^ be assumed for the axis of x, and let /q be the distance of the point of action from the projection of any particle of the cylinder upon the axis of z, Q the distance of the particle from the axis of ^. The conditions of the problem under consideration give /^ = (.r - |)= + 7;= + f ^ =/5 _ 2 .T? + i — (,>o) = ^nlciiil — (>l) ^V^i Go/ 131. ^YJlcn the imnt of action is in the interior void space of the"- shell, the constants of (5630) must have the same values as at the origin, where r vanishes. Hence, for this space, the constants are ^ = 0, The value of the potential in the interior void space is, therefore, constant, and there is no tendency to motion in any direction. 8 — 58 — 132. For an exterior imnt, the j^otential vanishes when r is infinite, while for a point at a great distance from the origin, its value is, hy § 128, the same as if the Avhole mass were concentrated at the origin. The value of the constants in this case are then i? = 0, yl = — m\ and the potential is r Any exterior iwint is, then, attracted ly a homogeneous spherical shell, precisely as if the luhole mass of the shell tvere concentrated upon its centre of gravity. ACTION AND REACTION OF A SURFACE OR INFINITELY THIN SHELL OF FINITE EXTENT. CHASLESIAN SHELL. 133. An infinitely thin shell may be reduced to either of its surfaces, upon which all its acting force may be concentrated, and the intensity of the action at each point of the surface will be the product of the corresponding intensity of the force of the shell, multiplied by the thickness of the shell, and the element of the surface must be substituted for the element of volume of the shell. Let then, do be the element of the surface, * N the exterior direction of the normal to the surface, Jc the concentrated intensity of action at any point of the surface, d\\i the elementary solid angle subtended by the element of the surface at the point of action ; the expression of the element of the surface is do =f^diiiseG^. — 59 — Hence — Ic dw = ^k da. The second member of this equation denotes the action exerted bj each element of the surface in a direction normal to the surface, and towards the interior of the surface. If, there- fore, the intensity of action is constant over the surface, the action normal to the sm^face is proportional for each element of the surface, to the solid angle subtended by the element, and the toted amount of the action^ normal to the surface, exerted hj any continuous extent of the surface, is jproportional to the v:hole solid angle sultcnded hj the houndary of the surface. 134. If the surface is a plane, the direction of the normal is invariable, and the total amount of normal action exerted by any portion of the plane is the same with the ^projection of the tvhole action of this portion of the plane upon the perpendicular to the plane, 2vhich is therefore proportional to the solid angle subtended by the portion of the plane at the point of action. 135. If the surface returns into itself so as to include a space, vMch is called a closed surface, and if the point of action is situated within the inclosed space, the tvhole angle suUended is the entire extent of four right angles ; tvhereas, if the point of action is exteiior to the closed surface, the tvhole angle vanishes ; hut it is tu'o right angles when the point is upon the surface. For, however the point of action is situated, if a line is drawn from it so as to cut the surface more than once, the successive angles which the line makes with the exterior normal, will be alternately obtuse and acute as the line cuts into the surface or out from it. The last angle, or that of which the vertex is most remote from the point of action will always be acute. The normal actions of two successive elements, therefore, upon the same line, and which subtend the same solid angle, are equal, but of — GO — opposite signs, so that tliey cancel each other's effect in the total sum of the normal forces. But if the point of action is without the surface, the first angle is obtuse upon each line, and as the last angle is acute, the whole number of intersections is even, and each normal elementary action is cancelled by another, and the whole sum vanishes. If the point of action is within the surface, the first angle is acute, if there is more than one ; and there are an odd number of intersections for every direction in which a line can be drawn ; for each direction, therefore, one, and only one, normal action remains uncancelled, which is proportional to the elemen- tary solid angle ; and the whole sum is that of the entire extent of four right angles. But, if the point of action is upon the surfiice, and a tangent plane to the surface is drawn through it ; every line which is drawn from the point upon the exterior side of the plane must cut the surface an even number of times, if it cuts at all, precisely as if it were drawn from an exterior point; but every line which is drawn upon the interior side of the plane cuts the surfiice, as if it w^ere drawn from an interior point ; the total sum, then, of the uncancelled elementarj^ solid angles includes those for all directions which are upon the inner side of the plane, that is, it is equal to two right angles. This elegant theorem, given by Gauss, is expressed analytically in the form X r 4:7t foi* a point interior to a closed surface, — = < 27t for a point upon the surface, f ' for an exterior point. 136. The expression (SOa) represents the component in the direction of the external normal to a surface, of the action upon the element of the surface of a mass /c concentrated at the point which, in that expression, was the point of action. The integral of this expression is the whole amount of such resolved action, and by — 61 — (G02i) its value is r — A: 7th when the mass /.; is interior to the surface, — / .,^ Jc =: — / Jc = -I — 2nk when the mass Jc is upon the surface, ^ (^0 when the mass h is exterior to the surface. Neither of these values depends upon the position of the acting mass further than it is interior or exterior to the surface or upon the surface. If, then, 3Ii = all the mass interior to the siurface, J[4 = all the mass upon the surface, Me = all the mass exterior to the surface ; tJie expression for the total action of the sum of all the masses upon a closed surface, resolved for each element in the direction of the external normal, is — in 31-2.71 M,r, and if all the masses are exterior to the surface, this sum vanishes. If the closed surface is one of the level surfaces of the system of hodies, this sum expresses the total attraction of the masses upon the surface. This impor- tant theorem is due to Gauss, and, independently to Chasles, in almost its full extent, as well as most of the following deductions. It is applicable, even if the surface have sharp angles, because the extent of surface occupied by such angles is zero. 137. If the closed surface is one of the level surfaces of a system of bodies, but not the outer boundary of a space in which the potentials constant, the potential must at each point, by § 67, increase in passing from the interior to the exterior or the reverse, so that in this case the sum (Glie) does not vanish. But the term of this sum, which depends upon the mass at the surface, may be neo-lected at will ; for the whole mass of a true geometrical surface is absolutely nothing. Hence, every level surface must inclose masses of — 62 — matter, unless it he the outer material houndary of a space in loldch the potential is constant. 138. When any masses lie upon the closed surface, the geo- metrical surface may, as Gauss observed, be arbitrarily assumed as being just exterior or interior to the masses, or passing through them. If, therefore, all the masses are so distributed upon a surface that it becomes itself a level surface, the potential is constant for all the inclosed space, and there is no tendency to motion throughout this space. 139. Around every point of maximum or minimum potential a level surface of infinitesimal dimensions may obviously be drawn ; and, therefore, every point of maximum or minimum potential must be itself a centre of action, and cannot be a void space. In an inclosed space, therefore, no point can be found for tvhich the value of the 'potential exceeds the limits of value tvhich are found upon the inclosing material surface; and in no point of unbounded space has the potential so great a value as its greatest value ujoon the exterior surface of the finite masses. This inference was drawn by Gauss. 140. In a system of bodies, of tvhich gravitation is the only force, there can be no point of absolute minimum p)otential. For if about a point of maximum or minimum potential, as a centre, an infinitesimal sphere is described, there can be no point within the sphere, either of maximum or minimum potential, wdth reference to the matter external to the sphere. But, with reference to the matter of the sphere itself, the centre must be a point of maximum potential, and, therefore, cannot be a point of minimum potential, with reference to the combined action of all the masses. ** This theorem is equally applicable to an aggregation of elec- tricity, all of wdiich is of the same kind, that is, wdiich is homogeneous when the point of action is assumed to be of the opposite kind of electricity. 141. If any extent of level surface is assumed at will as a — 63 — base, and if trajectories, like those of § 68, are drawn through each point of its perimeter, their nnion forms a canal. The same canal cuts a base, like the assumed base, from each level surface which it intersects. Of any canal, tJien, wJiich is not extended so far as to include portions of the attracting masses, the attractions upon all the hases are equal. For the whole amount of action, resolved in the direction of the external normal, at each point of action upon the closed surface, formed by the faces of the canal and the two terminating bases, vanishes, because there is no included mass. But there is no action perpendicular to the faces, that is, in the direction of the level sur- faces ; whereas the whole action upon the bases is normal to them. The actions uj)on one base are in the directions of its external normals, while those upon the other base are in the directions of the internal normals ; but these actions balance each other in the algebraic sum, and, therefore, their absolute values must be the same. This theorem belongs to Chasles, but the brief demonstra- tion is original. 142. In the follow^ing simple view of this whole subject, many of its propositions are condensed into a small compass. Each centre of action may be regarded as a fountain from which a stream is perpetually flowing in every direction, with an amount of discharge proportioned to the intensity of the action. The quantity which flows from each centre, for an instant, through any given elemen- tary surface, may easily be shown to be in exact proportion to the force with which the surface is attracted by this centre perpendicu- larly to itself and against the current ; and that which is true for each centre is also applicable to the combined action of all the centres. Upon a space, then, in which there is no spring, the amount which is flowing out must constantly be equal to that which is flowing in -, while from a space which contains springs, the amount which is discharged must exceed the inward flow by all which is — 64 — supplied by the fountains. These propositions are equivalent to those of § 136, and it may be shown by an easy argument that Laplace's equation, with its modification, is merely the same propo- sition applied to the element of space. By the additional hypothesis, that, to preserve the uniform flow of the stream, its density must decrease in each element of the stream with the distance from the origin, so as always to be inversely proportional to the distance from the centre, the potential represents the density of the combined streams, and the level surfaces become surfaces of equal density. The aggregate current of the combined streams is also equivalent to a single current in a direction perpendicular to the level surfaces, and having a velocity proportionate to the rate of decrease of density. But this is the well known law of the propagation of heat, w^lien there is no radiation, and hence arise the analogies between the level and isothermal surfaces, and the identity of the mathematical investi- gations of the attractions of bodies and of the propagation of heat which have been developed by Chasles. 143. If an infinitelij thin homogeneous shell is formed upon each level surface of a system of hodies, having at each point a thickness jyifoportional to the attraction at that point, the portion of cither of these shells, ivhich is included in a canal formed hy trajectories, hears the same ratio to the tvhole shell, ivhich the portion of another shell included in the same canal hears to that shell, provided there is no mass included hetween the shells. For if the bases of the canal are infinitely small, they must be reciprocally proportional to the intensities of the actions upon them, because the whole amount of action upon the different bases is the same. But the thicknesses of the shells are proportional to the intensities of action, and, therefore, the products of the bases multiplied by the thicknesses, or the volumes of the portions of shell included in the same canal, bear a constant ratio to each other. Since the ratios — 65 — are constant the infinitesimal volumes may be added together, and their sums, which are the volumes included in a finite canal, are in the same ratio, and these sums may even be extended so as to include the whole of each shell. Hence the volume of each portion is the same fractional part of the volume of the shell to which it belongs ; and, as each shell is homogeneous, the mass of each por- tion is the same fractional part of the mass of the whole shell. The conception of these shells, and the investigation of their acting and reacting properties was original with Chasles, and it will be con- venient, as it is appropriate, to designate them as Chaslesian shells. 144. The volume or mass of a Chaslesian shell has a simple ratio to the attracting mass included within it, dependent upon its own density and thickness. For each infinitesimal element of its volume or mass is proportional to the product of the element of the surface multiplied by the thickness of the shell, and the thickness at each point is proportional to the attraction at that point. The sum of all the elements, therefore, of either volume or mass, that is, the wdiole volume or mass, is proportional to the sum of all the attrac- tions upon the whole surface. But, by § 136, the sum of all the attractions upon the surface is proportional to the included mass, if there is no mass at the surface. If, then, II is the volume of the shell, Jc its density, h the modulus of its thickness, or the thickness which corre- sponds to the unit of attraction ; this ratio is included in the equation -^ — -^—471- 145. If a Chaslesian shell ivhich is ivholly external to the acting masses of the system is assumed to he itself the attracting mass ; 9 — GG — 1. The potential of the shell is constant for all interior points, there is no tendency to motion tvithin it, and its 02vn older surface is its level surface ; 2. Its external level surfaces are the same as those of the original masses of the system, and the attraction of the shell upon a point external to itself has the same direction as the cdtraction of the original masses. To demonstrate these propositions, let i2^ be the potential of the shell for any point, and £2 the potential of the original masses for each point of the shell ; the value of the element of the potential of the shell is Hence, In passing along the canal of the trajectories to another shell, the ratio of dji to f/ is, by § 143, constant, whence But dS2, _ led 11 ~ f dS2, Idn n ~ t^f d^i ^hdo Djs;i2 • and, therefore, diiD^f= — hdo D^S2 D^Ngo^ f =: — hdo D^il cosy , -p. d i2,, hli D-^ SI cos ^da The integral of this equation for the whole surface of the — 71 — Newtonian shell, and at the corresponding points there will be corresponding elements of volume. 151. The cotT'esponding elements of volume or mass of two corre- sponding Netvlonian shells are proportional to the volumes or masses of the shells. For if A-^, Ay, A^ are the semiaxes of the outer ellipsoidal surface of one shell, B^, By, B^ those of its inner ellipsoidal surface, o its volume, m its mass, and n the ratio of either axis of the inner surface, divided by the corresponding axis of the outer surface ; and if the same letters accented denote the same quantities for the corresponding shell, the construction of the shells gives for each axis B^ = nA^, X x' A A' J and n = ?/ ', and by differentiation, dx Ax dx' A'' The volumes and masses are by well-known theorems of geometry m ■=:Jco =^^ 71 k {A^ Ay A^ — B^ By B^) = i-7ik{l — n^)A,AyA,, m' = I/o'=iTik'{l — n')A',A'yA:. The ratios of the elements of volume and mass are, then, da dxdydz A^A,,A^ a da' dx'dy'dz! A'^A'^A ' A' /t" — TJ^ — dm hda ha m dm' kda' ko' m'' 152. If the Older surfaces of two corresponding' Newtonian shells have the same foci, their inner surfaces must also have the same foci. For if € ^ is the difference of the squares of the corresponding axes of the outer surfaces, the condition of the identity of foci gives the equations ,'^ = ^^ — A':^=:Al — A\^=..^^ — A7. HencC; for each axis, there is the equation Bl - B^ = n'^{A! - A7) = n'e\ so that the foci of tlie inner surfaces are also identical. 153. If the radius vector, from the centre of any point of an ellipsoid, is projected upon the radius vector of another ellipsoid ivhich has the same foci, and if the radius vector of the corresponding point of the second ellipsoid is projected upon that radius vector of the first ellipsoid, tvhich corresponds hi direction to the projeciion in the second ellipsoid, the two projections are inverselg p)roportional to the radii vectores upon tvhich they are p)rojected. For if Q is the radius vector of the first ellipsoid ujoon which the projection is made, and 'i, 1], L, are the coordinates of the extremity of q ; the equations of the corresponding points give, for each axis, whence r X x' A' A A 5 1 r — __ -x" X , and j/ the projection of r upon (^>', these projections are whence J- Q (> (> Q p' — Q 154. The difference of the squares of ilie radii vector es from the centre^ of two corresponding points upon the surface of two ellipsoids tvhich have the same foci, is equal to the difference of the squares of their semiaxes. For the equations of these surfaces are 2 /■'' NT-^ 1 ^_ 1 The difference of the squares of two corresponding radii vectores for j)oints at the surface, is ,^ - r- = 2^{.^-.-) = 2, [.r = (l - f )] 155. The distance of any point upon the surface of an ellipsoid, from a point upon the surface of another ellipsoid tvhich has the same foci, is equal to the distance of the two corresponding points of the ellipsoids from each other. For if 10 — 74 — / is the distance of the point of which cc, ?/, z^ are the coordinates, from the point of which i', i/, t' , are the coordinates, and f the distance of the corresponding points ; the values of these distances become, by (Tog.io) and (73.24), '2 9//,/ p=r'+e-'^u *> 21 '2 1 2 i) I I 9 1 /9 o ' ' y2 whence 156. The external level mrf aces of an ellipsoidal Chaslesiau shell are those of ellipsoids ivhich have the same foci ivith the outer surface of the Chaslesiau shell. For if SI, is the potential of the given shell for any point of the external ellipsoidal surface of the same foci, and 11[ the constant value of the potential of the corresponding Chaslesian shell, constructed upon the external ellipsoidal surface, for any internal point, and, therefore, for any point of the surface of the given shell ; the equations (72^) and (74i2) give o The value of £2, is, therefore, constant for all points of the surface of the external ellipsoid, so that this is one of the level surfaces of the given shell. 157. The attractions of two corresponding Neiiionian shells, which have — 75 — ilie same foci, upon an external point, have ihe same direction, and are propor- iional to the masses of the shells. For the infinitely thin shell, this proposition is a simple corollary from (742,5). But the finite shells can be subdivided into corresponding infinitesimal shells, and the masses of the corresponding elementary shells will be proportional to the masses of their respective finite shells. The attractions of the corresponding elementary shells upon an external point, there- fore, coincide in direction, and are proportional to the masses of the shells; and, therefore, the components of all the corresponding attractions have the same common ratio, and coincide in direction. But the components of all the elementary attractions constitute the attractions of the finite shells themselves. Several special cases of this theorem were first given by Maclaurin, but the general form was first demonstrated by Laplace, and afterwards more rigorously by Legendre, and it includes the case in ivliich the inner surfaces are reduced to the centred point, and the shells hecome ellipsoids, having the same foci. 158. The attraction of any Chaslesian shell upon a point at its surface is, from its construction, perpendicular to the surflice, and proportional to the thickness of the shell at that point. The attrac- tion upon the whole surflice is, therefore, proportional to the mass of the surface, which corresponds to § 136. Hence, if dN is the thickness at any point, and p the perpendicular from the centre upon the tangent plane at that point, the attraction of the ellipsoidal Chaslesian shell at the point is 4 Tt Z" dN = 4 TT Z: dr cos ^. ■=. 4>'t/:— rcos ;• p A 7 ^^r = 4:71 /Cp—r-, — 76 — The component of this action in the direction of the axis of x is 47rA"»-i— cos . If, moreover, the equation of the ellipsoid is the general theory of contact gives N D,L pD,L cos — — X— ^(QZ) 2:,{xB,L) Hence, 2x 2 D.L^'4, ^-UL—^ - 2,xD,L = 2Z,^,= 'l, -■A N px and the attraction in the direction of the axis of x of the ellipsoidal Chasle- sian shell upon a point at its surface is 'inkp^x—T^. 159. The attraction of an ellipsoidal Chaslesian shell upon any external point is obtained by describing the corresponding Chaslesian shell, for which this point is upon the outer surface, and the attractions of the two shells for this point have the same direc- tion, and are proportional to their masses ; so that the attractions in any direction are proportional to the masses. If, then, the accented letters refer to the outer shell, the attraction of the inner — 77 — shell is IGO. The condition that the outer surface of the exterior shell passes through the attracted point, is expressed by the equation This is an equation of the third degree when it is reduced to its simplest form. But there are two other surfaces which can be drawn through the given point, and which depend for their defini- tion upon the solution of the same equation. They are two hyperboloids, both of which have the same foci with the outer surface of the inner shell, one of which is a bipartite, and the other an unparted hyperboloid. For each of the hyperboloids £^ is negative, and its absolute value, independent of its sign, is contained, in the case of the unparted hyperboloid, between the squares of the mean and least axes of the given ellipsoid, and, in the case of the biparted hyperboloid, between the squares of the mean and greatest axes. 161. The points in ivhich all the ellipsoids, ivhich have the same foci, are cut hj the common intersection of the two hjperholoids ivhich have the same foci, are corresponding points. For if £'^ is the value of — £^ for either hyperboloid, the equation of the hyperboloid for the points of intersection with the ellipsoid is 'A'. — i!' If the equation (77;) of the ellipsoid is subtracted from this equation, the remainder divided by e^ -f- ^'^ is — 78 — r.'-^ in which /, y\ and z are accented, in order not to interfere with the notation which has been adopted for the corresponding points, and which gives for each axis X x' x! The substitution of these equations in (78i) reduces it to 7^ = 0; the 2:)roduct of which by fc'^, added to (TGq), is 1, ---AliAi-^^)— ^Ai — ^^ which expresses that the point [x,y,z) is upon the surface of the hyperboloid, and, therefore, all the corresponding points are upon the surfaces of both hyperboloids. 1G2. The hf/perholoids and elKjJSoids ivhich have the same foci, inter- sect each other loerpendicularly . The conditions that two surfaces of which the equations are Z = 0, and ^=0, intersect each other perpendicularly is expressed algebraically by the equation for each point of the line of intersection. But for the hyperboloids of equation (7728) ^^d the ellipsoid of equation (76e) this condition becomes Ai{Ai. — £ -) ' which is the same with the equation already given in (78io). This — 79 — same demonstration may be applied to the condition of the perpen- dicuhirity of the hyperboloids, if Al is diminished by fc'^, and t^ is changed into the difference of the squares of the semiaxes of the two hyperboloids. 163. It follows from these two theorems, which are derived from Chasles, that each normal transversal to the ellipsoidal surfaces of level is the line of intersection of two hfjperholoids tvhich have the same foci. 164. The lines of intersection of these three surfaces are, upon each surface, the lines of greatest and least curvature, for they are a special case of the theorem demonstrated geometrically by Dupin, that the intersections of three surfaces ivhich cut each other at right angles at and infinitely near their common point of intersection, are their lines of greatest and least curvature at this point. To demonstrate this theorem, let the three normals to the three surfaces at the common point of intersection be assumed for the axes of rectangular coordinates, and let be the equation of the surface, which is perpendicular to the axis of X. This condition gives for either of the other two axes in which equation r^•,^, and z may be mutually interchanged, except that the same axial letter must not be repeated in the equation. Those equations satisfy of themselves the condition (T825) of per- pendicularity of these surfaces at the point of intersection. But the intersection of any two of these surfaces coincides w^ith the axis which is the intersection of their tangent planes for an infinitesimal distance, and the two surfaces are perpendicular to each other for this distance. Hence, each pair of surfaces gives an equation of the form — 80 — which is reduced by (792o) to The other surfaces give the corresponding equations The sum of the products obtained by multiplying the first of these equations by D^L^, the second by — D^L^, and the third by DyLy is 1DM.BMBl.L^^^. and the corresponding similar equations are obtained by advancing each letter to the following letter of the series, rr, y, z, and x. But the factors D^L^, ByLy, and D.L^, are not zero, and, therefore, these equations may be reduced to which are the well-known conditions that the directions of the axes ofx,?/, and s respectively coincide w^itli those of the lines of greatest and least curvature of the three surfaces at the origin. 165. The remarkable relations of these surfaces might be still further extended, and if it were worth wdiile to investigate the attractions of masses of infinite extent, it might be shown that upon each series of orthogonal transversal surfaces, Chaslesian shells of infinite extent might be constructed. The level surfaces of these shells would be the orthogonal transversal surfaces of the same series, while their orthogonal transversal surfaces would be the level surfaces of the original Chaslesian shells and the other series of orthogonal transversal surfaces. 1G6. To investigate the attraction of an ellipsoid upon an external point, it may be supposed to be divided into an infinite — 81 — series of elementary Cliaslesiaii shells. Let then A^, Ay, yL, be the semiaxes of the ellipsoid, ((_,, a,j, a., those of the outer surface of either of the elementary Chaslesian shells, and let a , a„ a, -t'i_t ^"ly _tl-. If, moreover, .r, y, .t, are the coordinates of the attracted point, A'_^, Ay, A[, are the semiaxes of the ellipsoid, which has the same foci with the given ellipsoid, and whose surface passes through the attracted point, u^, ciy, a., the semiaxes of the ellipsoidal surface, corre- sponding to the outer surflice of the Chaslesian shell, and passing through the point of action, JE^ = A'j} — A1,, and £ -- ;r = a^' — «:^ = (1/ — A^_^n-; the values of E and £ are the roots of the equations S ^' =-^ ''' — 1 The attraction of the Chaslesian shell upon the external point in the direction of the axis of .r is by {11 2) ^^'^•^ ~zrry~T • ~:~ — -tTi kx , 3 , , . — , a^/UyCi', a J. a'^^a'ijCi'. ' n in which the value of p is, by equations (TOg^ig), given in the form 1 _ UL _ ^ -' /^ — [2-(xA.X)]^ X- ^ ^ ~1 iyl ~T a 11 — 82 — The differential of (SUi) after it is multiplied by ir is ^vlience by (8I31) 11 ^ (In = -,'cdt . V - This value reduces the attraction of the shell in the direction of the axis of .f to — 4 71 lex ' ,3 ; , t at, = — J ;t /c.v ^[{Ai^ey{AlJ^e^){Ai^,^)y The integral of this expression is the attraction of the whole ellipsoid. The limits of integration correspond to the values of t, for one of which the shell is evanescent, and for the other its surftice coincides wdth the surface of the ellipsoid. But, when the shell vanishes, n is zero, and e is infinite ; and when its outer surface coincides with that of the ellipsoid, u is unity, and t becomes E. Hence, the expression for the attraction of the ellipsoid in the direction of the axis of x is, if M is the mass of the ellipsoid, and /rits mean density, \ J . 2Kj^(Al-\-e')s/[{Al + e^iAl-i-s^)(Al-^e-)-] By advancing each letter in the series x, fj, £■, and x to the follow- ing, the corresponding expressions are obtained for the attractions in the directions of the other two axes. 1G7. By the substitution of h\ = Al + .\ — 83 — the equation (8224) becomes r> _ ^^ f ^ 1G8. By the substitution of ?r, = ^-, and 6,, the equation (883) becomes j^ f^ ?jMx r kill KA,.,)u^ sJ[{Ai-^ul{Al — At))iAl-\-uliAl — A^)y Avhich formula, Avith tran.- = Al — Al + {Al — Al)f^mHl>^{Al — Al)QOfiH, (Al + ,'')£n''ip=Al — Al, d.-c^ =■ — 2(^1 — Al) cosec^(j cos (fd(f ; Avliicli, substituted in (82.24), reduce the expression for the attraction in the direction of the axis of x, when the eUipsoid is homogeneous, to the form -I-^ •^ n 71*- [* sin -ff sec ^.^ o Mx i' , , ,s J 6 i-^ X — -^ -; (A ,. — A'.) -' sm - i t/ or, if i^ = the attraction is ^ sin - i ^ ' ' The same substitution gives the attractions parallel to the other axes in the forms Dy il = Py I sin ^ 9 sec ^ ^ , ^ A-^^ = P^ Ttan^c/nseci? ' But the differential of the logarithm of (80.21) is cot(^Z>.<^ = cote/), and, therefore, D.^ =z tan () cot 9^', — 85 — D^ ( tan f/ cos (.^ ) = sec ^ ^ cos (3 — sin ^ t^ sec (3 = sect3(sec^9cos^t3 — sin-t?) = sec (3 sec ^9 — sec t^ sin ^ t^ (sec - (/ -)- 1 ) = sec (3 sec ^9- — sec t^ sin ^/( tan ^f/ -|- sin-(/) = sect3 -|- cos ^ / sec (^ tan -9) — sect? sin- (3 =^ cos^ -|- cos^ /sec (3 tan -91, D i sin q- cos u sec t3 ) =. -^4>0'^^^cos(ir) __ cos (y cot qr sec ^ g tan d — tan ^ sin y f^' ^ ' ■' ' ^ sin? sini = cos^^) sec'^t3 — sect'sin^c^ = — cos ^ /sin ^f/; sec'^^ 4" (1 — sin-(/ sin^/) sec'^^ — sec (3 sin ^f/) 9 • • 9 q < I A /t sin'"^\ = — cos-^sm-ff sec'^d -I- sece ( 1 1 ' \ sin-// = — cos-ism^'msec'^ti + -^^-.(cos-t3 — cos^O ■* ' sni -i^ ' = — COS - /sin ^ (p sec'^ t^ — cot"/ sec t' -|- cosec" / cos t3 . These equations reduce the attractions to the forms rk f^ 7) r rsec^ecosi? — sec^ " • t^ / • wl Dyll = P^J [ ^.^,. sec-«i>^(sni9cos9- sect^) J ° = P^(icosec^2/^^«?J — cosec^ /?)=, — sec^/sincjf>cossec6)), i>^/2 = sec^/P^ /* [i>^(tanff.cost3) — cos(3] =: sec^/P^- (tan c/^ cos — ^.fF») • 170. The following values are derived from (8323_24) and (8I15); cos2= ^ — sin 2 .1; — J; A'J — A^' Ai + E-'~ aT~' Al — Al A'^ — A:/ Ai-^E-' A';:" SG cos-- = ,, I j^.., = -TT,, Sin ■' I COS" I Al — Al^Al — A^ A X — A J A^ — A^~ Al — Al A? — A';' Al — Al — A:/ — Ar The equations of the attractions give^ by means of these values, that of Panel (8I15), j^^^^ =:, 2^^D,^I2 = sec2/PsinfZ'secf/'sec0(cos2r9 — cos-0) SM , M This simple equation is due to Legendre, and the first of the two following equations which are obtained by the same process of reduction. 171. By putting X = . f ^ ^ - \ g; tlie attractions may assume the form DJl^ — ^MzB^^ L. in which the differentiations, relatively to A\, A^, and Al, are performed without regard to the changes oi E, dependent upon the formula (Sl^j). — 87 — 172. The e(|uatioii (Sl^.j may, by meaiLs ui" the e(|Liatioiis (8^2-7) 'je written in the form x'--^1/Hqg'' + 5-.sec--'/' = (Al — Al)co^ec'-'P, or by the substitution of the value of from (8824), a:- + ^-sec2(Z* , y^ 1 Ai — Al ' A:.cos^-0-\- A fsin - (p — A l ^m-(p' 173. Whcu the attracted ^^oiut is iqmn the surface of the ellipsoid, E vanishes, and the value of becomes cos'f' = -^. Ax 174. VChen the attracted point is tvithiii the ellipsoid, the Newtonian shell, of which the outer surface is that of the ellipsoid, and the inner surface passes through the point, exerts no action upon the point, and the attraction is reduced to that of an ellipsoid similar to the given ellipsoid, and of ivhich the surf ace passes through the attracted point. 175. When the density of the ellipsoid varies in its interior, in such a way that each of its component Chaslesian shells is homo- geneous, Jc is a function of f, and after its substitution (8294) may be integrated. 176. When the ellipsoid is a homogeneous ohlate ellipsoid of revolution the various formulae become ^'^+/ + .^-' + ^-'tan^* = (.4|-^!)(l+cot^ =: Al — Alx ^ I),J2 ^Fgf sin^c^ = iP^(2^/' — sin2). 88 — ■ 177. When the ellipsoid is an homogeneous prolcde ellipsoid of^revolu- iion, the formulas become A = Al. i = l^f, T = ^, D^ll 1= Px / sin^g)secy ^ := P;i' [log tan {{n -\- i ) — sin «/»] , D,j£l = P u I sin^f/'sec^f/) ' == i ^U [sin f/' sec^ f/> — log tan ( ] ^t: + ^ ^Z^)] , P-wi = P.5' f sin^cj}sec^9 = I Pz [sin *fJ sec^ ^ — log tan {\n -\- \ f/>)] . ATTRACTION OF A SPHEROID. LEGENDRe's AND LAPLACE's FUNCTIONS. 178. The investigation of the attraction of a spheroid is greatly facilitated by the introduction of certain functions which were first conceived and investigated by Legend re, but which became so fruitfid in their more general form, given in the subse- quent researches of Laplace, that they are usually designated by the name of the latter geometer. A method will be pursued in their development and discussion which is similar in some respects to that given by Jacobi. — 80 — 170. Lot IIzzzi COS w -\~ /sin fj" Qo^i] , and if any power of II, denoted hy n, is developed in a series of terms arranged according to the cosines of the multiples of iy, let any one of the terms be denoted by in which [;;^-] denotes the number of accents of ^7^. The required power has then the form CO IP = ^,„ {("' ^^ir^ cos m r^ ) . 00 180. The value of His not changed by reversing the sign of rj, and, therefore, the series remains unchanged by this reversal of sign, which gives or ^^-'"^ = ± 4^1"^ = (— l)'"t/>lr^; in which the upper sign corresponds to the even values of m, and the lower sign to the odd values of m. The equation (80n) nif^y also be Avritten 1 181. The integral of the product of (8O22) by cosniij is, by a well known theorem f {II"co8mri) = 27i /'« cPIr^ . The derivative of this equation, relatively to (p, reduced by the condition D.R=^ — sin if) -f- 1 cos (p costj , 12 — 00 — becomes 9 n i"'D^ '^P\r'^ =^n I ^B" ~ ^ cos ni i] ( — sin (p -j- z cos ip cos ?y )] ^=n I [ff"~\ — sinf/)cosm>;-|-^?*cos(/(cos(m-|-l)iy-]-cos(?« — l)'i))] ; whence, by (SOgr), 182. The derivative of (SOgo), relatively to i], reduced by the condition D^ Hzzzz — ^ sin (p sin i] becomes CO 2 « //" - 1 sin f/) sin ij = 2 :r„, (m /'" c/>lrJ sin ?;^^ >; ) . The integral of the product of this equation by sin;«j^ is in sin (/> / ( 7/"'" - ^ sin i; sin ?/? i^ ) = 2 tt ?« z '" [;"i , • / 7/ or in sin rp / [//'" - ^ ( l cos (w^ — 1 ) i; — i- cos (?« 4- 1 ) ^y )] = 2 tt ?;^ i'" [;"] ; '' which becomes by (SOg;) ?;« ^^[f'] = 1 ?^ sin (p ( ^/'L"!-/] + *^lri+i^J) . 183. The equation (892;) may assume the form j lIP~\Gos^'Comri-{-lism(pj{cos{m-\-l)ij-^(cos(m — l))']=27ii'" (p^J^\ which, reduced by (8%-), gives ^P\l"l, 4- -1 sinf/.(c/>i;'r-/] — f/^[r_+/^). — 91 — 184. The remainder, if (OOg) is subtracted from the product of (9O24), multiplied by cotf/, is n cos (f 4^\;"Jl^^ 4- n sin o) [;"i = ^^Z>,o,^(sin'"+i9)^^[r+^^)- 185. It will be found convenient here and elsewhere to adopt the functional notation 1 rh=C(-\og.:c)\ which gives, by a familiar formula, or by simple integration by parts, when h is positive, and Jc an integer, which is less than // -)- 1, 1 rh = h (h — 1) (/^ — 2) (h — /c + 1) f{— log.r)^-^- = h{h — i){h — 2) (h — k-}~ i)r{h — Jc), — 02 — and /,(/,_ l)(/,_2)....(A_Z-+l) = ^,^. When h is an integer, and h the next smaller integer, this formnla becomes 1.2.3 Ji=rh. With this notation, Taylor's theorem assumes the form 186. The equations (Oli^) and (Olso) give, by successive sub- stitutions in each other, and the use of the preceding notation, (— l)'"r(« — m) sin'" 9^ 0f;"] = (— iy''r{n — m') B':^-''' (sin"'> 4>l:"^) ; in which negative differentiation must be interpreted to be integra- tion ; in the former equation, when n is negative, m -\-n-\-l and m -\- 11 -\- 1 must be positive ; while, in the latter equation, n, n — m -f- 1, and n — m -\- 1 must all be positive. When n is posi- tive, but n — m -\- 1, and n — m -\- 1 are negative, the equation to be substituted for (92i5) is I\m — n — l) r{m' — n — \) ' which equation is also to be used when n is negative. When n and n — m -\- 1 are positive, but n — m -\- 1 is negative, the combina- tion of (92i5) and (9293) gives, by representing by n\ the greatest integer contained in n -f- 1, (_ 1) "'-"' r('M — m) sin '" (f (//;'0 /■(,/ _ n — 1) i);:;;^'" (sin"'V/,ri;f;"'J) /•(« — n') r{m' — n—\) ' When «, n -\- dI -\- 1, and n -J- r^i -\- 1 are all negative, the — 9: equation to be substituted for (92i3) is I\—\~n — v\)^\\V"^ l\—\—n — iv!) •=°«'? siii"''g' When n and ?i -[~ ^'^'+ 1 ^^'^ negative, but m -j- >^ + 1 is posi- tive, the combination of (92i3) and (903) gives, by representing by n, the greatest integer contained in — 1 — ?«, D\ There are pecuUar considerations which simplify the investiga- tions, when n is integral, whether it be positive or negative ; and these are the cases to which most of the subsequent investigations are limited. 187. By reducing 711 or vi to zero, the equations of the pre- ceding section give, for positive values of «, ^(—AY' rnr{m-n-\) p ^^ V ) r{n — 7/) r(n' — 71 — l)sm"' cf J cos ^ "' and for negative values of n ^" ~ r(— 1— «)sm™(yJcos^ " — r{—l—n) ^^^^ ^|^cos6^n 188. When n is zero, it is easily seen that 11^=1 = ] or the series is finite for ])ositlve integral values of n, and contains onlg n -\-\ terms. 190. The substitution of the preceding equation in (9l8) gives c/^w — x^in(f(p\l'Zx^ = (1 sinf/))'"^fr„T^ = (ising;)'^0o = ^sm"9. which equation, substituted in (93i,3), gives fp = (-1Y =^"i>;os^(^[r^sin» = ^^^L,(- sin^/O" ; =z vzz}zl j)n-m ( ginS^.yt . 2T(7i — m) sin'" cp '"'"!' ^ '^ ' bg loliich the coefficients of the development are oUained ivhen n is a positive integer. 191. When n is the negative of iinitg, the equation (89.27) givQs But the value of //gives - 05 — cos rp — ^ sin rp cos ?j // cos (jp -|- ^ sin q> cos // cos '^ cp -j- sin - cp cos - // /sinrjpcos?/ ■ cosqp 1 — sin^^sin^/y ' cos ^ /; -[- cos "^ g) sin ^ j; i-rsinfjfcos// i^/sin g^cos?/ . cos g Z),^ tan »;• 1 -|- sin (f sin // 1 — sin cp sin ij~^ 1 -\- cos ^ qt^ tan ^ // ' the inteo-ral of which is C I T -1 1 -j- sin (T- sin ?7 i , r n/ j. \ / 77 = — l-^ iO^ r^ — . ■ - 4- tan ^~ ^J ( cos q> tan 7; ) J yrl -• °1 — smqfsin// ' ^ ' '^ Hence, by passing to the limits ^-1 = 1, When n and m vanish, equation (90gi) becomes 0^ =: cos 9J ^*[^'i — (;^sinc/. + £^)'^'^'I — secg, ^^sin"-> V^-"^^ IjSinf/;-t- ^.^^^ )i-n, which, when m = « — 1 , — 96 — is reduced to CIJ[ 11-11 The successive substitution of 1, 2, d, &c., for n, gives, by means of (91,2) r sin " qj r(2}i) , . N„ The substitution of this value in (9821) gives, by (942o), and, therefore, for all values of m less than n -\- 1, The equation (9l8) gives, when m — n vanishes, by (969), whence, by (9O24) From this equation the successive values of *?'[!.\j may be deter- mined by successive substitution of 1, 2, 3, &c., for n, or they may be determined by the equation derived from (9I20) a,nd (969), cos (i — 97 — The remaining coefficients, in wliicli m is greater than n, are then to be determined by the equation derived from (92i3) ; - " r{7n — n) """^ <> sin > • 193. In order to apply the preceding investigations to the problem of attraction, it is requisite to introduce the form of polar coordinates, of which zenith distance and azimuth is the familiar instance. For this purpose let the following notation be adojDted : ^yis the angle which a line/ makes with the axis, ^f is the angle which a plane, drawn through the axis, parallel to/, makes with the primitive plane. The distance /, between two points, of which the radii vectores are r and i), is given by the equation /2 =r r 2 + (^) 2 _ 2 r () (cos c/., cos ^^ + sin ^, sin c/)^ cos ((1, — ^p) ) = ^''+^>' — 2()rcos^. Hence the notation i^sl — 1, Hf = cosgiy -f- 2 sin c/ycos {ij — ^y) , gives ■^fllf = +/COS f/ly + Z/Sin 9)yC0S (i] — ^y) ^ /'cosf/'^ — (^cosf/) -|- /cosiy(rsin9^cos^^ — f)sin^ cos^ ) -|- /sin ri (r sin c^, sin (3^ — ^ sin 9) sin ^ ) in which the upper sign is to be used when r is greater than ^, and the lower sign when r is less than (>. But it folio WS; from § 191, that lit "i- ' ' ill which the uj^jper sign corresponds to r, greater than q, and the lower sign to r, less than (), and the series represented by the fourth member corresponds to the former of these two cases, while the series represented by the last member corresponds to the latter case. 194. If, in the development of the preceding series, Q„ is the coefficient of -^, and Q[, is that of -^, ; the series become The values of the coefficient in these series are determined by the equations 27r H'h If the additional notation is adopted, corresponding to (SOy) Hr = [/•]. + 2 1. {i- [r]L'"^ cos;;^ (;; — d,)) , 1 the values of the coefficients become 1 CO «. = ['■].. [',']-<« + .. + 2 ^„,((- 1)'" [r]L"" [(Oi^','..+.co,s™ {i\, - ^)). — 9'J — Hence, by (96,s) and (93ii;), «.= q:= Da,[o],.+2|„,( ^<"-^;>y;'+"'> [r];r> [o]ir' cos,« (.^^ - cv)) =['a.[co..+2i4^;^;«''i>s,,;r],.i>r.,,^[-o,.cos„<^-t^^ 195. The equation (45^) gives for the value of the potential, Hence, by the notation the potential becomes oo rr 00 S2 — :z-^z=::^( v -/■"). ^ With the notation of (0823.27) ^^^^ (Q'J'io) these values become do ^()^du' d o = o 2 sin cppd(fpd^pdQ, The first form of 12 in (99^) is to be used for all values of (^) less than r, and the second form for all values of o greater than r. If, then, k is supposed to vanish for all points of space in which there is no attracting mass, the limits of integration for the value of U^ must include all the attracting mass for which q is less than ;•, while those for U', must include all the attracting mass for which o is f^reater than r. — 100 — 196. By substituting in (992i_23) the values of Q,^ given in (992) the resulting values of U^ and U^ have the same form with Qn so far as the elements of the direction of r are involved ; so that the value of the term of U,^ which depends upon the angle m^^ ^^^ the form \f]l''^ ( JiL'"^ cos W^ ^r + -^ir^ sill m &r) , in which A\'"'' and i?[r^ are independent of the form of the body, and the number of such constants included in the most general value of £/"„ is 2 ?2 -]- 1. 197. It is expedient to introduce, at this point of the discus- sion, some important properties of Legendre's functions. The following theorem, given by PoissoN, is of especial use in ficilitating their investigation. If Np denotes any function of the elements of direction of (), and if, after the performance of the integration expressed in the second member of the folloiving equation, q is made equal to r, tvhich condition is intended to he denoted hj the subsequent parenthesis, the second member zvill be reduced to the first member, that is, 4 T ^ To demonstrate this theorem, it is to be observed that all the elements of the integral vanish, except those for which that is, for which /=o, ", =0, If, then, 1] denotes the angle which the plane of r{> makes with any assumed fixed plane passing through r. — 101 — the integral becomes, by (9 Tie), 1 r r ()•■' — n^-) Xp iVv r r r (r^ — ry-) i^m] 27r 198. The equation (9Tio) gives, by means of the first form of (98,.,), r'-'f = -p + rfDJ = -/' (j, + r a)) which substituted in (IOO20) reduces it to 4- *=f-(^-/,"'-''-')=f-^"i-'' by adopting the notation 47r in which it must be observed that when n is zero it must be retained in the written expression to avoid confusion. It ma}- also be remarked that, from the comparison of the forms of (99i2) and (IOI21), the most general form of JS'l"^ is the same uith that given in § 19G for Z7„. 199. If the given function is such that, for every value of?/ different from n iVTJ = o, — 102 the equations (101i7_2i) give J 4.. ( TV^t"]"! — A^f"] ^ tJ ll) The theorems expressed in the last two equations are of fundamental importance, and tvere given hij Laplace. 200. The theorem (IO27), not being limited to any special direction of r is true for all directions ; and, therefore, the most general form may be substituted for Q,,, which can be obtained by combining all its special values in any linear function. Any such general form would be the same with that of Nf^-, and if it is denoted for distinction by M\f^, the theorem (102^) assumes the more general form given hy Laplace, 47r ^ 201. In considering the attraction of a spheroid upon an external point, luhich is so remote that r is greater than ang value of (^ let u be the value of {) for the surfiice of the spheroid, and CO 1^ J P the function which is denoted by the second member of this equa- tion being developed in the form of a series of terms of Legendre's functions, by means of (IOI21). The equation (992i) becomes, by means of (IO25), and the potential is 103 202. If the point is so remote that the squares of the Hnear dimensions of the body may be neglected in comparison with the square of the distance of the attracted point, it has been shown in § 128 that the attraction is the same as if the body were condensed upon its centre of gravity. In this case, therefore, if the origin is assumed to be the centre of gravity, the potential becomes, as in (56^), III all cases, then, in ivhicli tJie origin is the centre of gravity, this erjiia- tion gives for an external point tvhich is so remote that r is greater than {>, Uq = m, n — !?! _i ^ ( ^^ VM^ 203. A homogeneous ellipsoid can always le found, of ivhich the potenticd, for any external point, developed in the form (lOogo), t^ill l>e iden- tical tvith this expression in its two first terms. To demonstrate this proposition, and develop the mode of investigating the ellipsoid in a given case, it may be observed that, if the centre of the ellipsoid coincides with the centre of gravity of the given spheroid, and if the mass of the ellipsoid is the same with that of the spheroid, the potential of the ellipsoid, for an external point, has the form (lOSso) with the same first term. The difficulty of the demonstration and investigation is thus reduced to the consideration of the second — 104 — term. The general form of this term is, by (OO^) and (100c), i?[2] = A (cos'c/), — J ) + smf/),cos^,(^cos^, + ^'sin^,) + Bmhp,{Ccos2^+ 6''sm2^,) = — ^A-\- Acos^l -f- Clcos^l — ^^^^y) -[- -^COS^COS^ -f~ ^ COS^COS^ -J- J 6' COS^^COSy = :^, ( (7, (cos^^ — I) + ^.cos^cos'^) ; in which last form the arbitrary constants C^, Oy, C^, B^, By, and B. are introduced, for the sake of symmetry, and in which the six constants are only equal to five, by reason of the equation JS'_gCOS^^= 1. In the especial case of Q2, these constants become, for the axis of :r, ^zrz Scos^cos?, 6^ = 2 cos X ? and similarly for the axes of y and ^. The equation (TGg) of the homogeneous ellipsoid, of which the axes are the given rectangular axes, gives, for the surface of this ellipsoid, 1 ,^ cos^^ If, therefore, K is the density of the ellipsoid, the equation ( 10227) becomes, in this case, u 00 ^ 1 — 105 — and hence, by (1002), 47r *j 1/; ' To obtain the value of TJ^^ it must be observed that, by (104i-), 47r because, from the symmetrical form of the ellipsoid, the value of ?( is not altered by changing either of the angles upon which it depends into its supplement, while the sign of B^^ is reversed. The remaining terms of the integral contained in (1062) have the form 47r 47r 4 tt r (it'O:) = f r (it' cos^ i\ = I J (?r^cos2f/v.). ^ ^ But, by the equations cos^ = sine/) cos ^^,, ( 10425) becomes cos"" =sincf) sini^ , 1 Qos'^cfp I s'ln^q^pCos^dp . sin ^ (jryj sin - /9p u ^ A 'i "T" A l "^ A I cos- (9,, sin"^ On '% Ai "I" Al "I \^^; Ai Al } ' > by putting rr=K«' + yC0s2^p); 14 — lOG — 1 , 1 a =1: -— -^ -- , A'^ ' Ay 1 A- T ^ 1 Z* = -7^, — «. The integral (lOoij) is, therefore, 47r 2;r tt ^ 0*^ •i TT TT __ 3 r r cos^psinyp ^ J Gp J cf^ {a -\-bcos'^ cpp) 2 ' But J* eo-i"^ cf p sin cfp ^ — cos'qrp cpp {a -\- b cos ' (/ p) - 3rt {a -\- ^ cos"- f/p) - ' TT r cos ^ g-p sin g-p ^ 2 2 Az J Cf^ (a-\-b(jos''Cfp)' oa{a-\-b)- 3 a ' Jd.a j-20p2a J2dct'^b\ P 2 cos 26p ^ r 1 (a' — b') tan ^ ^^,tant-i(^tan<5j, i TT J 00^ A. A, These values, with that of the mass of the ellij)soid, reduce (lOGg) and (105o) to 4^7r "J lb U, = -1 71 KA^A,A,:^^ ^AUo8''^-iAt) lb — 107 — = ^\m^^(AUo^'l-iA'^ If the axes of the ellipsoid are not those of :r,^, 2', but of /,?/',/, this expression, by means of the equation, COS_^' = COS^COS^^/ -f- COS COS ^/-|- cos^cos"/, becomes in which ^^2 = /o "^^4^^' V^^^^ ~~ + ^^'cos^cos^J; This value becomes identical, therefore, with that of (1048), if /^/^ -^ fi ■^ dm 204. If the potential and its component functions for the ellipsoid are denoted by the letter e Avritten beneath them, the potential of the spheroid for an external point, for tvhich r is greater than o, becomes n = n + f,, [( u^ - ^„)r-<"+^G . e 3 e 205. A transformation of coordinates, which is the reverse of that by which the equations (lOTn) were obtained from the reference of the ellipsoid to the axes of the spheroid, would bring the equa- tion (104i9) to the form (lOTie). From the forms of the expression it is obvious that this transformation is identical with that by which the general equation of the second degree in space is referred to the — 108 — axes of the surface. Hence, if S^r, Sy,, and S^r, are the three roots of the equation ( c:- s) ( (-;- >s') ( c:— s) + 2 kj^:k- ^. \bi\ c:-^)-\ = o , they are the squares of the semiaxes of the ellipsoid. But it must be observed that the mass of the ellipsoid, being the same with that of the spheroid, gives the equation Cf Cf Cf / ^ '" \" The condition (104i4), however, shows that the values of C^, O^, and C[, in (1048), may be increased or decreased by the same quan- tit}", without changing the value of (lOlg)- The values of C'J, C'J, and C^^ may, in like manner, be increased or decreased by the same quantity, which change will produce the opj)Osite effect uj)on the roots of (IO83), until, at length, they may satisfy the equation (lOSg). This common increase or decrease of all the roots of (IO83) corre- sponds to the performance of the same operation upon the squares of the semiaxes of the ellipsoid, that is, to a change of the ellipsoid, given by (IO83) into another ellipsoid, w^hich has the same foci and the required mass. The change of mass is, however, more simply accomplished by an increase or decrease of the density of the ellip- soid ; and, in this view of the case, it is requisite that the value of the density be determined by equation (IO89). 206. If the point is without the spheroid, but near its surface, it is generally necessary to combine the forms of the potential given in (99ic). Thus, with the notation J= the integral for all directions of u greater than r, I = the integral for all directions of u less than r, — 109 — whence ^J -ip *J -ip ty Tp tlie value of the potential may be expressed in the form in which r But it may be observed that, by (lOOa)? whence, by putting ^^' — V^-'h^ J PL'"-'' and using U,^ in the signification of (QOia), the potential assumes the form f-X^.-J^v*?-)) """ n \ .. J/ 4- 1 I 207. A similar investigation may be extended to the ellipsoid of§204, andif S2' is the value of 12 of (IOT24) — 110 — the value of the potential for a imint tvhich is near the siuface of the spheroid may assume the form si = n'-i.f\(v,,.-i-,:)Q.). 208. If the form of the spheroid differs hut little from an ellipsoid which has the same foci tvith the preceding ellipsoid, and if it has a constant density for all that portion for tvhich {> is greater than r, a combination of tivo homogeneous ellipsoids may he suhstituted for the single ellipsoid, hoth of tvhich have the same foci, ivhile one coincides very nearly tvith the spheroid in form and density throughout the portion exterior to r ; and the other, hdng much smaller, has the requisite positive or negative density to give the alge- hraic sum of the masses of the two ellipsoids equal to that of the spheroid. The combination of the two ellipsoids upon any external point is the same with that of the single ellipsoid, and the larger of the two may be substituted for it in the values of Fin (llOg). If, in determining the values of V for the spheroid or the ellip- soid from ( 10922), ^^ is supposed, for every direction in w^hich the solid is contained within the sphere, of which radius is r, not to refer to the surface of the solid, but to coincide with r, the value of F van- ishes for any such direction, and it becomes a continuous function, of which the derivatives are discontinuous. The equation (IOI22) is applicable to such a function, for the argument by which it was established was independent of this condition. With this modifica- tion, therefore, the accent may be omitted in the integral sign of (1092;) or (llOg), and the limits of integration extended to every possible direction, and the result may be simplified by means of (101^2). In the present case, in which Jc is constant, equation (IO922) becomes ^<" = ,7+3 ( r'^- ) + ;r:r-, {^^ - ''\ , — Ill — whence If, then, it is assumed that ^p = u — r, /^^ = u — r, e the binomial theorem gives and if w is changed into — [u -\- 1) «-2^" ^ )— -Al\a-i)rm^~') -'l')- These values, substituted in (III2), give V>^—Vn — ^ r ^ / n^^ + 2) , (-i)'»r(n-3+»o x ,„ ^,1 ^ Kp ;-'"L/'mr'»-iV/'(^^ + 3 — wO "T" r(« — 2) )\"p—"i^)\ 1 ^ [_!_ / A» + 2) , (-irr(n-3 + »0 X _ -1 "T" 7 '" L / '« '•'" ~ ' \^(« + 3 — m) ~f" /•(« — 2) / '^•^P "P ^ J • This value may be substituted in (IIO3), and the result reduced by means of (IOI22). 209. If the spheroid is not very different from a sjjJiere, and if the difference in form hetiveen it and the larger of the two combined ellipsoids is so small that, in consideration of the large divisors, the terms of (III20) mag he neglected, in which m is greater than 3, (III20) i^ reduced to V,n-V,,. = {2n-\-\)W„ c if W 1 7«/^2 y2\ 1 Ji / 3 ,/3\ . — 112 — and, by (110,o), But the value of the potential, derived from (IIO3), becomes in this case by (III28) and (IOI22), 477 n = £1' - i ((2« + 1)/ ( n; §„)) ^ pj _ 4 ,^ 2^^ Ti^['o ^ n' _ w, ■ — (y . so that, in iJiis case, the form (IOT24) ^'^ ap'pUccible to ever?/ external point. This conclusion, and the mode of investigation, includes Poisson's analysis of the spheroid, which differs but little from a sphere by which it was suggested. 210. The formula (IO724) gives, for the attraction in the direc- tion of the radius vector, the expression e 8 e Hence, the equation is obtained ^ 1 e ^ ' e 3 e which, by (1082), is reduced to or 00 e 3 6 211. If the spheroid is homogeneous, having the same density with the ellipsoid, the equation (lOlgi) gives — 113 — ^p"* — ::r]-^K^^y n + 3"^ Bj assuming, then, the values ' e e 1 /„_Lf! „_1_Q\ TT- 1 V' w + 3'^-^p :p >*' '^'■" — „+3^-^ :'• >'' o' __ o i2 • e the equation (11228) becomes 212. 7/^ ^/^e attracted point is upon the surface of the spheroid, the preceding equation becomes, if P.^ is the potential at the surface of the spheroid, D,n.', + Ul', = - 271 Kr 3:^ v^^. 213. If the spheroid differs so little from the ellipsoid that the square of the distance between the surfaces of these two solids may he ner/lectcd, the notation e gives ^p" — ■' p Up — -^p Up- 214. If, moreover, the ellipsoid differs so little from a concentric sphere, that the product of the difference between the radius vector of the ellipsoid and the radius of sphere, multiplied by the distance between the surfaces of the ellipsoid and the spheroid, may be neg- lected, the preceding equation is reduced to 15 — 114 and (llon) becomes In this last form, the sum of the terms in the second member is extended to include the whole series, because the first terms which vanish in the exact formula, may become sensible in the approxi- mate form. But, V — ^ ;/f"^ • and, therefore, if II is the radius of the sphere, Ai2J + ^,.Q; = -2.rA^i?y,. 215. If, again, £1^ = the potential of the ellipsoid at its surface, e S2q =z the potential of the ellipsoid at the surface of the e spheroid, e e e the general equations e « give Since the second members of these equations are midtiplied by — 115 — y^, the values of the other factors may be reduced to those which belong to the sphere. Hence, ^p« becomes a constant quantity, and, therefore, for all values of n except zero, in which case, e " e and the above values become e which give The sum of this equation and (114i2) is, by (llog) and (1142o), 216. If fhc dlijisoidis itself the sphere, the equation (SSg) gives e e T) r) I _L o^ — 2 jiXR ' which, substituted in (lloi;), gives T) n _i_J_r) — i-r KR This is the equation given by Laplace for a spheroid which differs but little from a sphere, and is the fundamental theorem of his investigations upon this subject. — 116 — 217. If the aitr acted foint is ivitJiin the spheroid, and at such a dis- tance from the surface that r is less than the value of u, the formula for the potential is, by § 195. in which 4 TT »■ ' V'=f'' f^. J ih Jp Q ' It may also be shown by the method of §§ 208-209, that this same formula is applicable, if tlie point is quite near the surface, and if the spheroid differs so little from a sphere that the square of the difference may he neglected. 218. The important discussions in regard to the convergency of the series, derived from Legendre's functions, are deferred, on account of their great length, to the volumes which will be devoted to the application of the Analytic Mechanics. IV. ELASTICITY. 219. The laws by which the elementary forces oi cohesion and affinity vary with the mutual distance and direction of the particles and atoms are undetermined ; and, therefore, the delicate inquiries involved in the constitution and crystallization of bodies are not yet subject to the control of geometry. But it is sufficiently apparent that these forces are insensible at sensible distances, and that there are peculiar laws of mechanical action corresponding to the three states — 117 — of f/asses, liquids, and solids. The peculiaritv of these states consists, prmcipally, in the faciUty with which the particles can be moved relatively to each other, and in the phenomena which arise from such motion, but especially in those of the disruption of solid bodies. As long, however, as the relative positions of the particles are so little disturbed that they return to their initial state when the dis- turbing cause is removed, the precise law of molecular action is not required for the investigation of the small changes which the consti- tution of the body undergoes, and which are treated as phenomena of elasticitij. 220. To analyze the changes of form of a system of material points which constitute a body, let u be the distance by which a point, of which the coordinates are .r, y, and 0, is moved from its initial position, A the increment of a function for another point of the body which is near the former point, IJ the distance of the second point from the former point ; the notation of (42,) gives P ^pcos^, ^p. =PxD^ u^ ^PyDyU^ -\-P-^ A ?r Hence, if P =P + ^'Jp, Ap ~ P ' £ is the linear expansion of the body in the direction of ^; ; and its value is given by the equation (i+.)-(^y=^.a 2 p — 118 — = ^,[(1 + i>,!gcos^ + D,tt,conl + A«.cosi']'. If, then, the reciprocal of\-\-^is laid off from the origin upon a line draivn parallel to p, its cxtremitf/ ivill he upon the ellipsoid, of ivhich the equation is 1 = ^., [(1 + D^ur) X + D.u^y + D^ji^iJ. 221. The expansions or contractions which corresjiond to the axes of this ellipsoid may be called the p>fincipal expansions and con- tractions, and one of these is a maximum, another is a minimum, and the third is a maximum in some directions and a minimum in others. If the ellipsoid is referred to its axes, the expression for the expansion is, if e^;? ^y? and e. are the values of g for the axes, SO that for these directions the values o^u^, Uy, u,, are such that (1 + I),u,)D,^u, + n^Uyil + DyU,) + BjLDytL = 0. (1 + e.^y = (1 + B^u^y + (i>,.g^ + (i>,,gi 222. The notation P iJTives cosf/) = -ZJcos^cos^ ) ; sin^ cp = 1 — cos^fj) == JS-.cos^^.^^cos^^ — [^.(cos^cos-^)] = -^^^cosycos^ — cosCcoSy I ; '\12 119 — (1 -|- f)^sin^f/j = —.^^(pcos^cos^ — ycosjcos^j =^ ^J (^cos^i>^ -)- COS y By -f- coh^^dMuzCos^ — i^yCosl^) 1 |2 in wliicli the derivatives are only applicable to ii^, Uy, and il. Hence, if the reciprocal of the square root of {\ -f- *) sinf/) is laid off from the origin, upon a line drawn parallel to p, its extremity is upon the surface of the fourth degree, of ivhich the equation is 223. When the axes are those of the ellipsoid of § 221, and the disturbance is such that for each axis the equations (llSjg) and (II829) become (l-|-e)^sin^9) = ^^|coSy cos^(Z>,2f. — DyU,^ = ^^[cOS^COS^(fc, — £^)J . 224. To determine the rotative effects of the disturbance about the axes, let P and ip^^ = the projection of the angle (p upon a plane perpendicular to the direction of ^. Hence — 120 — , / , N pi I),ti,cosP -\- Z),,M,cosi; 4- (1 -f- Am,) cosP 225. If the axis of;?; is perpendicular tojK»j the equations are P — Xfy—P tan (y, -h 1/' j - 1 _^ ^^,^,^ ^ A^^tanx/; ' 226. If the axes and conditions are those of § 223, the equa- tion (I2O3) becomes tan(f/), + 1/',) = \'\_'^^ll tani/^,. 227. The whole expansion or contraction of the body at any time, is derived from the consideration that, by the definition of e in § 220, any very minute portion of the body which is originally a sphere, becomes, in the disturbed state, an ellipsoid similar to that of §221. If, then, ^ 1= the expansion of the body ; the sphere of which the radius is i, becomes an ellipsoid, of which the axes are i{l -\- ^^), i{l -\- ty), i{l -^ aS), and, therefore, its vol- ume becomes and l + « = (l + e,)(l + g(l + a,). 228. When the disturbance is so small that the squares of the expansions may be neglected, which is the ordinary case of elas- — 121 — ticity, the equation (119,3) becomes « = — .r [cos 2 J A?'.. + COS ^ COS ^' {D,jlL -\- A?^)J =: :r., (cos I a) ^.. (cos ^! u)j . 229. If, then, the reciprocal of the square root of the linear expamion in any direction is laid off from the origin upon that direction, as the radius vector of a surface, the residting surface is a surface of the second degree, of which the equation is or l=:^,(xi>J^,(x«J. 230. If the axes are those of the principal expansions and contractions, the formula for expansion becomes £ = ^.(cos2j.fc^); and the equations of § 221 become Djly-\- DyU^=^ X '■'^x' 231. If the principal expansions and contractions are all of the same name, that is, if all are expansions, or if all are contrac- tions, the surface of § 229 is an ellipsoid. But, in other cases, in which neither of the principal expansions is zero, the surface is the combination of two hyperboloids, of which one is one-parted, and the other is bi-parted. Both these hyperboloids have the same axes and the same asymptotic conical surface ; and the asymptotic conical surface, corresponding to the directions, in which there is neither expansion nor contraction, divides the directions in which the solid is expanded from those in which it is contracted. IG — 122 — If one of the principal expansions is zero, the surface is reduced to a cjh'nder ; and if two of the principal expansions are zero, the surface is reduced to two parallel planes. 232-. In the present case, the formula of § 227, for the expan- sion of the soHd, is reduced to 233. The formula (I2O9) for the rotation about the axis of x becomes, tan {(p, + i/') = t'-in if' + -^ = (1 -|- Z>-?f, — J^yif,/) tani/^ — D^u,,iaii^ i\i -\- D^iL, (p.. = li^^y^'z — A?^) + li^y^iz + A«y)cos 2 1/^ -|- 1{D^ 11^ — Dy iiy) sin 2 1^ = n. + "^x cos 2 ( J/; — 1],) ; in which T^ sin 2 ij^ = 1 (A ?fz — ^y "y) • 234. The maximum rotation about a: corresponds, then, to W — ''U and is c/)^ = i/, -[- t^ ; and the minimum rotation corresponds to r = Vx + ^, is and Jf^ is the mean rotation. When the maximum and minimum rotations have opposite signs, there are two intermediate rotations — 123 — which vanish, corresponding to cos2(i/; — ry,)=— ^. 235. There are simiLar fonniilre for rotations above the axes of y and z, and the combinations of the mean rotations give a great- est mean rotation, represented by the direction of which is determined by the equations represented by Tl IT, cos,=-. 236. If the axes are those of § 230, the equations of § 233 become f;, = 77-, + M A v.. — D, li,) sin 2 w . 237. When the disturbance is such that, for each of the prin- cipal axes, there is the equation the equations of the preceding section become 77, = 77=0, 9, = 1 (Z>,«, — DyU^) sin 2 w ; so that, in this case, there is compression vjithout any mean rotation. 238. When the disturbance is such that for each of the princi- pal axes — 124 — the equations for compression and rotation becomes so that, in this case, there is rofation ivitJiout compression. All the preceding investigations upon the internal changes pro- duced by the disturbance of the form of a body are taken from Cauchy. 239. The elastic force which is developed by any small dis- turbance of the internal condition of a body is proportional to the amount of disturbance, and has, therefore, the same general form with that of the disturbance itself But the special discussion of the relative values of the coefficients involves the consideration of the laws of equilibrium, and must be reserved to a subsequent chapter. V. MODIFYING FORCES. 240. Among the forces of nature, those which produce the equations of condition deserve peculiar consideration. Being merely conditional, they do not augment or decrease the power of a system, but merely modify its direction and distribution. They may, therefore, be called modifying forces ; and may be divided into two classes of siaiionary and moving. 241. Stationary modifying forces are perpendicular to fixed sur- faces or lines, and constitute the action by which certain material points of a system are restrained to move upon those surfaces or lines. A force of this nature, being perpendicular to the motion of — 125 — its point of application, does not increase or diminish the total power of the system, but modifies its elements of direction. Thus the equation of condition, X = 0, between the coordinates of a point, involves the idea of a force, acting in the direction i\'^of a normal to the surface represented by this equation. When it is combined with its multiplier, it is equiva- lent, by (STie), (^Ts), and (543i), to a modifying force, of which the magnitude is "o 242. This force may be decomposed into three forces, wdiich are parallel to three rectangular axes, either of which is represented by ?.v'(aX)cosf, while the point of application moves through the elementary arc ds, its advance in the direction of the axis of x is ^5008*. The amount of power added to the system, by the component force in the direction of the axis of .r, is Xt/5v/(n-^)cosfcos*, and there is a consequent increase or diminution of force in this direction. But the mutual perpendicularity of iVand s is expressed by the equation -Z^(cosfcos^) = 0. The whole augmentation of power arising from the three com- ponents is, therefore, >.^5v/(n^)-^.(cos^cos^) = 0, — 126 — which agrees with the fundamental conception of a stationary modifying force, and illustrates its mode of action. 243. Moving modifying forces are ^^ei'pendicular to moving sur- faces, which surfaces are themselves portions of the moving system, and the points of application are restrained to move upon these surfaces. In this case, the motion of each point of application may be decomposed into two parts, of which one part is perpendicular, and the other is parallel to the moving surfaces. The modifying force has the same relation to the motion which is perpendicular to it, which has been already discussed in reference to the stationary surface ; put by its relation to the other component of the motion, it communicates power to the point of application, or the reverse. But the power which is thus communicated to the point is abstracted from the surface, and through it from the other por- tions of the system ; and, therefore, the whole amount of power of the system is neither increased or decreased. Although for the pur- poses of theoretical speculation, it is convenient to regard the sur- face and the point of application as parts of one system, it is often the case in the useful arts that this transfer of power is of the highest practical importance, and is the basis of the theory of the turbine wheel. In a rigid system of bodies, these forces constitute the honcls of union. — 127 — CHAPTER VI. EQUILIBRIUM OF TRANSLATION. 244. The conditions io ivliich amj comhinaiion of forces must he sub- ject, in order tlicij may not tend to produce translation in the system of material points to ivhich they are applied, are readily investigated. It follows immediately from §§ 18 and 20, and with the notation of those sections, that the algebraic condition that the system has no tendency to move in the direction of ^; is ^;w2il\cos/=:0. But each term «?l^lCOSj , is the projection of the force m^F-^ upon the direction of p, and, therefore, if the algehrcdc sum of the projections of all the forces upon any direction vanishes, there is no tendency to translation in that direction. 245. It also follows from the combination of translations, given in § 23, that if there is no tendency to translation in two different directions, luhich are not pfarallel, there is no tendency to translation in the plane of these two directions ; and if there is no tendency to translation in three directions, tvhich are not in the same place, there is no tendency to translation in any direction. By means of rectangular axes the algebraic conditions, which are necessary and sufficient to produce equilibrium in respect to translation, are combined in the formula ^.[-;(;;^i/^icos;)]2=0. This formula is independent of the situation of the points of the — 128 — system, except so far as the elements of position are implicitly con- tained in the expressions of the forces and their directions ; it would remain unchanged, therefore, if all the points were condensed into one, without any variation of the magnitude and direction of the forces. The conditions of equilibrium are, then, the same as if all the forces were applied at a single point. 246. If one of the points of the system were subject to the condition of being confined to a fixed surface or line, the conditions of equilibrium of translation w^ould simply be reduced to the condi- tion that the 7'esultant of all the other forces umild he ijerpendiciilar to this surface or line, and the modifying force hj tvhich the ijoint ivas restrained ivoiUd he equal and opposite to this residtant. If a point of the system was absolutely fixed, or if three differ- ent points were restrained to move upon three fixed surfaces, there would, in general, he no possihility of translation, hut the residtant of all the forces applied to the system ivould he equal and opposite to that of the modi- fying forces hy which the points ivere confined. 247. The theory of the equilibrium of a point is wholly included in that of its translation. But since every system is a mere combination of points, the complete theory of equilibrium can easily be evolved from that of translation. This mode, however, of arriving at the conditions of equilibrium is neither luminous nor instructive. 248. The conditions of the equilibrium of translation of a sys- tem, which is free from the action of all stationary modifying forces, may assume the form, that each force is equal and oppodte to the result- ant of all the other forces. If, then, there are only two forces, they must be equal and opj^o- site ; and if there are three forces, they must all lie in the same plane, and be represented by the sides of a triangle formed by three lines which have the same directions with the forces ; so that each — 129 — ' force mmt he proportional to the sine of the angle included hetiveen the other two forces. Whatever are the forces, if we were to start from a point, and proceed in the direction of either of the forces, through a distance proportional to the intensity of that force, and proceed again, in the same way, from the point at wdiich we arrived in tiie direction of another force ; and so on, proceeding successively from each new station in the direction of the next force, through a distance proportional to that force, the course w^ould finally termi- nate at the original point of its commencement. CHAPTER YIL EQUILIBRIUM OF ROTATION. 249. The conditions to ivhich a system of forces must he siihject, in order that it may not tend to produce rotation about a point or an axis, are directly deduced from §§ 84 and 88, and are simply, that the resultant moment of all the forces, tvith reference to the point or the projection of this resultant moment upon the axis, must vanish. 250. When there is an equilibrium of rotation about a point, the resultant of the forces may not vanish, in which case there is not an equiUbrium of translation. About any other point, there- fore, which is not situated in the line drawn parallel to the resultant through this point, there is not, by § 100, an equilibrium of rota- tion ; although there is an equilibrium of rotation about every point of that line. In order, then, that there may he an equilibrium of rotation about all 17 — 130 — points of sj)ace, or even about three points not in the same straight line, there mnst he an equilibrium of translation as tvell as of rotation. 2-51. In the same way, it apj)ears, that if there is an equilibrium of rotation about parallel axes lying in the same plane, there is an ecquilibrium of translation in the direction perpendicular to the plane ; and if there is equilibrium of rotation cd)out parallel axes tvhich are not in the same plane, there is an equilibrimn of translation in everg dircciion except that of the parallel axes. 252. If there is a fixed point in a system, it is necessary and sufficient for the eqidlibrimn of rotcdion that the resxdtant moment for this point should be nothing ; and, in this case, the resxdtant moment vanishes for every point of the strcdght line tvhich is drawn through the fixed point par- allel to the resultant, and cdso for every axis tvhich is in the same plane tvith this strcdght line. 253. If there are two fixed points in a system, it is necessary and sufficient for the cqidlibrium of rotation thcd the moment of the forces shoidd vanish for the line 2vhich joins the two points. 254. If all the forces are parallel and equal, there is, by § 99, combined with § 250, a line parallel to the common direction of the forces for which the resultant moment vanishes. If the common direction of the forces is assumed for that of the axis of z, the moment of the force acting upon a particle dm, with reference to an axis drawn parallel to that of ^ at the distance a, from the plane of yz, is [x — a)Fdm, and the whole moment of the system is r {x — a)F=Ff (x — a). The condition therefore that the moment vanishes for this axis is / [x — «) = ; — isl- and the plane which is thus drawn at the distance a from the plane of yz, includes, by § 127, the centre of gravity. Hence, the axis, for ivMcli the resultant moment of the ixirallel, and equal forces acting upon a system vanishes, passes through the centre of gravity ; and if the system has an equilibrium of rotation, and if there is afxcdpoint in it, the centre of gravity must he in the straigM line tvhich is draivn through the fixed point in the common direction of the forces ; or, if there is a fixed axis, the centre of gravity must lie in the plane ivhich includes this axis and the direction of the forces. It is also apparent that, if the centre of gravity is advanced heyond the fixed point or axis in the direction of the forces, the equilibrium is stable ; but if the centre of gravity is not so far advanced as the fixed point or axis, the equilibrium is unstable. The ordinary case of gravitation at the surface of the earth, in which its variation in intensity and deviation from parallelism is insensible for the small system of bodies discussed in the usual investigations of mechanics, is the flimiliar type of this s^oecies of force. 255. In the motions of translation and rotation there is no motion of the parts of the system among themselves. There is no change, therefore, in the mutual distance of the origin and point of application of each of the forces which arise from the action of the parts of the system upon each other. The origin, regarded as a point of application of the same force, acting in the opposite direc- tion, moves just as far in the direction of the force as the actual point of application ; so that such a force acts precisely as a moving, modifying force, and has no tendency to affect the equilibrium of translation or rotation. All the forces, therefore, between the different parts of the system may be neglected in determining the conditions of the equi- librium of translation or rotation. This mutual relation of the origin and point of application of the force, by wdiich either may be regarded, at pleasure, as beirtg — 132 — the origin or the point of apjDlication, by a, simple reversal of the direction of the force without any change of its intensity, is com- monly expressed by the proposition that action and reaction are equal. CHAPTER VIII. EQUILIBRIUM OF EQUAL AND PARALLEL FORCES. MAXIMA AND MINIMA OF THE POTENTIAL. 256. In order to give precision to the modes of expression, and have the benefit of well-known terms and forms of speech, the force considered in this chapter, is assumed to be the typical force of gravitation at the surface of the earth, acting within a space small enough to admit of the neglect of its variation of intensity and deviation from parallelism. The level surfaces of this force are horizontal planes, and the potential decreases uniformlg ivith the increase of height ahove the earth's surface. 257. Let the three rectangular axes be so assumed that the plane of xz is horizontal, the axis of g, the upward vertical, that of X, the northern horizontal line, and that of z, the western horizontal line. If, then, g is the intensity of the force of gravity, y G the distance of the centre of gravity from the origin, and — 133 — £2q the value which the potential would assume, if all the points were in the plane of .-r.* ; the actual value of the potential is, 1jy the property of the centre of gravity, = £2,— f G^ = S2, — mG^. Hence the potential is a maximum, ivhen the height of the centre of gravitij is a minimum, and such a position of the system corresponds, hj § 62, to that of stable equilibrium ; hut the 'potential is a minimum, when the height of the centre of gravity is a maximum, and such a position corresponds to that of unstahle equilihriwn. 258. Since the direction of gravity is the same for all the points of the system, there cannot he an equilihnum of translation, unless there are stcdionary modifying forces, the resultant of tvhich must he exactly cqucd to the whole weight of the system, and have a verticcd, upvydrd direc- tion. 259. The resultant moment of all the forces of gravity van- ishes for the centre of gravity ; and, therefore, the resultant moment of all the stationary modifying forces must vanish for the same point. 260. If there is but one modifying force in the system, it must he vertically directed upimrds, have an intensity equal to the lohole iveight of the system, and its line of action must pass through the centre of gravity. 261. If there are but two stationary modifying forces, they must lie in a, common plane, tvhich is vertical, and includes the centre of gravity, their resultant must have an upivard direction, and he equal to the iveigM of the system, and they must he reciproccdly proportional to the dis- tances of their directions from the centre of gravity. This last condition is involved in the necessity that the resultant moment must vanish for the centre of gravity. — 134 — 262. If the intensity of the force of gravity were to be increased or diminished, the conditions of the position of equiUb- rium would not be changed, but intensity of the modifying forces would be proportionally increased or diminished. Even if the force of gravity were to be made negative, that is, if the direction of its action were to be reversed, the conditions of the position of equilib- rium would still remain unchanged, provided that the modifying forces were of such a nature that the direction of their action would also be reversed ; but, in this case, the position of stable equilibrium becomes that of unstable equilibrium and the o^Dposite. This rever- sal of the direction of gravity is relatively accomplished by the rotation of the whole system about a horizontal axis. II. THE FUNICULAR AND THE CATENARY. 263. When the points of application of a system of forces are united by a single continuous chord which is destitute of mass, the polygon, which is formed in the situation of equilibrium, is called a funicular. The general conditions of such a system involve a mere repetition of the principles of equilibrium ; and the present discus- sion is limited to the case, in Avhich the points of application are masses acted upon by gravity. 264. When there is but one fixed point to the sj^stem which may, Avithout any essential loss of generality, be assumed to be either extremity of the chord, in every position of cquilihrimn the cJiord must he vertical. But if the idea of the incompressible rod is supposed to be included in that of the inextensible chord, each portion of the chord included between two successive masses may be assumed to have a — 135 — vertical direction, either upwards or downwards ; so that, if n is the number of masses, 2 " is the number of positions of equihbrium, all of these positions, except that one in which every portion of the cord is directed downwards, involves an element of instability, and must, therefore, be regarded as abwlidely iimtahle. The tension of each jyoiiion of the chord is, in ever?/ case, equal to that of all the weight which it has to sustain ; that is, to tlie sum of all the subsequent masses ivhich lie upon the fortion of the chord not attached to the point of suspension. 2 Go. When there are two fixed points, the whole included chord must hang in the same vertical plane w^ith these two points. The tensions of the various portions of the chord rejjresent modifying forces ; and the surfaces at which these forces act are those of spheres, all the centres of which are movable, except those of the two fixed points. In the position of equilibrium, however, all the centres become stationary, and the conditions of equilibrium of each mass or portion of the chord admit of independent discussion. The forces which act upon each mass are gravity and the ten- sions of the two portions of chord upon each side. The horizontal projections of these two tensions must, therefore, be equal and oppo- site in order to balance each other ; so that the horizontal projection of the tension of the cJiord is invariable throughout its zvhole length, and equal to the horizontal projection of the sustaining force of each of the fixed points. The algebraic sum of the iipimrd vetiical projections of the tensions at the two extremities of any portion of the cJwrd must be equal to the iveight of all the intermediate masses in order to support them against the force of gravity. 266. These two conditions are necessary and sufficient to produce an equilibrium of translation in any portion of the chord, and, therefore, of the whole chord. The condition of the eqiiihb- — 136 — rinm of rotation of each portion of the chord, although included in the preceding conditions, is an interesting and nseful modification of them. With reference to the centre of gravity of the masses of each portion of the chord, the moment of the gravity of the masses is zero, and therefore the moment of the tensions applied at the extremities must also vanish. But the directions of these tensions are not parallel, and therefore their lines of tension produced must meet at a point, at which both the tensions may be regarded as applied without affecting their tendency to produce rotation. At this new point of application they may be combined into a result- ant, which is vertical, because the horizontal projections of the ten- sions are equal and opposite. This resultant has the same tendency to produce rotation with the tensions themselves, and therefore it must pass through the point for which this tendency vanishes, that is, through the centre of gravity of the masses. The point of meeting, therefore, of the lines of extreme tension of amj I'^ortion of a chord is in the same vertical ivith the centre of gravity of the intermediate masses. 2G7. If the two extremities of any portion of the chord are in the same horizontal line, the equal horizontal projections of the extreme tensions are exactly opposed, and therefore the moments of the vertical projections of these tensions must be equal with reference to the centre of gravity. The vertical iwojections of the extreme tensions of any iiortion of the chord, of ivhich the extremities are tijjon the same horizontal line, are, then, reciprocally proportional to their distances from the vertical draimi through the centre of gravity of the inter- mediate masses. 268. Since the horizontal projection of the tension of the chord is the same throughout its whole extent, no portion of the chord can become vertical. If any portion of the chord is hori- zontal, the vertical projection of its tension vanishes, and, therefore, — 137 — the vertical projection of the chord at any other point is equal to the sum of the weights of all the masses intermediate between this point and the horizontal portion. If then T is the tension of the chord at any point, and if the axis of x is horizontal, and that of y vertical, directed upwards, so that T^ is the horizontal projection of T, and Ty its vertical projection ; and if s is the arc of the chord at any point, and m the sum of all the masses included between the point and the horizontal portion of the chord ; the following equations express the preceding conditions : T^oK z=z T., Tcos; = T,= m tan^ = m The inclination of the chord to the horizon, therefore, increases as the distance recedes from the horizontal portion. If the chord has actually no horizontal portion, the preceding- equations are still applicable by assuming for in, such a value as would be required to correspond to the vertical tension of any given portion of the chord. 269. If, in proceeding from the horizontal portion in either direction, the chord is everywhere ascending or descending, its hori- zontal direction must also be away from the extremity of the hori- zontal portion to which it is attached so as to form a portion of a convex polygon, which cannot be intersected more than once by any vertical line. Such a position of the chord corresponds to that 18 — 138 — of the perfectly stable state, or to that of the most unstable state ; and each state is always possible. If, in proceeding from the horizontal portion, the direction of motion changes from ascent to descent, or the reverse, the horizon- tal direction must be reversed at the same time, and so that the subsequent portion of the chord will form an arc of a polygon which will include the preceding portion within its concavity, and the con- cavities of both portions will be turned the same way. 270. The difference of equation (ISTis) applied to two differ- ent portions of the chord gives the following eqviation between the intermediate masses, the horizontal tension, and the directions of tension at the two points, m. — m sin (^' — %) 2\ cos ^' cos ^ 271. If the masses are infinite in number, and arranged in unbroken continuity so as to form the chord itself, the curve is called the catenary. In this case, if h is the weight of an unit of length of the chord, the mass of an element is dm =. lids ; and if ^ = the radius of curvature, the equation (ISSjg), applied to the extremities of the element, gives, for the equation of the catenary. ^ — X :-^sec- X If A=: T. this equation becomes i>^s = ^) =z ^sec 2 J — 139 — 272, If the chord is of iinform ihichicss and dcnsitf/ throughout its length, Jc and A are constant, and the integral of (13 831) is s = iltan^, to which no constant is added, because the arc is supposed to be measured from the point at which it is horizontal. 273. The curve of the uniform chord is easily referred to rec- tangular coordinates, for the equations D^^g =1 D^ s sin ^^= A sin -^ sec Dt X = Bi s cos ^ = ^ sec ^ : 2 5 a:? give, by integration, and determining the constants, so that the ori- gin may be at the point of horizontality, ^ = ^(gec^ — 1), X := J.logtan i (|- 71 — ^) . These equations give, by elimination and the use of the nota- tion of potential functions, Sin^ = tanJ = -^, 274. The vertical tension of the uniform chord is and the whole tension is !r= T^sec^=T^CoB^=T,(l_ + l) = T^^{. — 140 — 275. If the chord ivcre required to he of such a variable thichicss as to assume a given form of curve, the law of this variable thickness is given by the equation The vertical tension is -^ V ^^^ ^ "^ ^^^ 2T J and the whole tension is 5^=I;sec^ 276. If the thickness of the chord loere required to le ijroportional to its tension, so that T the following equations are successively obtained by easy transfor- mations Sin -^, =. tan ' = tan ^., J = log sec I =: log Cos ^, ^) =: ^sec -^ = ^ Cos -g = c^, rr= T.sec^= 7;sec J=. T,Qos^=T,cy. 277. i^ //^e thichiess he such as to give an uniform horizontal distri- hution of the weight, that is, such a distribution that the weight of each portion of the chord is proportional to its horizontal projection, the equations are ^_ y,.cos^ — 141 — D^s = () = Cscc\, and the curve is a parabola, of which the transverse axis is vertical. 278. If the chord were compressible and extensible, it would be compelled to assume that thickness, in which it would have the requisite tension ; and the form of the curve would, with this condi- tion, be the same as if it were incompressible and inextensible. Thus, if F denotes the function which expresses the given law of the relation of the thickness to the tension, so that rp -^ /7T ? ■'- X ^ X the form of the curve is given by the equations 1 D'^.8 = Q cos'^^F{sec'^,;) n sin Dix — 1 cos J i'' (sec \*)' 279. If the chord or any portion of it is confined to a given surface, the resultant of gravity and the tension of the chord on each point must be normal to the surface, and is balanced by the modifying force by which the point is fixed to the surface. If, then, the tangent plane to the surface is, at each point, assumed as the plane of x ij ; if the axis of x' is horizontal, and that of y directed upwards, and if (^/ is the radius of curvature, at this point, of the projection of the chord upon this plane ; the curve and tension may be determined by means of the equa- tions 142 — T T^. ^ ^- silly, cos J,', ^'cos^^-cos^/' Z-'sin*, cos^/COSy', ^ ^ ' D,T^=zJc cos Ir COS I, = k COS l^k D,?j , 280. The pressure upon the surface is determined by the con- sideration that it must exactly balance the tendency of each point of the chord to move in the direction of the normal to the surface. But the tendency of the tension to move any point of the chord in any direction, as that of ^;, is In the case of the direction JV of the normal to the surface, this expression becomes, because s is perpendicular to JV, in which q" is the radius of curvature of the projection of the chord upon the common plane of the normal to the surface, and the tangent the chord. Hence the pressure sustained by the surface in the direction of the normal is i? = ^^+^"COSl. 281. If the chord is destitute of imght iqwn any portion of the siir- — 143 — face, {)' hecomcs infinite, and the curve is that of the shortest line tvhlch can he drawn tipon the smface. The tension, in this case, is constiint, and the pressure upon the surface becomes T Q 282. In the case of a cylinder, of ivhieh the axis is vertical, the equations become y' — y ^^7 T so that the curve is the same when it is developed ivith the cylinder into a plane, ivJiich it assumes ivhen it hangs freely. 283. In the case of a surface of revolution about a vertical axis and a chord of uniform thicJcness, the equations become T=Jc{y+y,), ^ sin ^, cos ^,' in which the angle which y makes with y' is determined by the meridian curve of the given surface, the plane oixz passes through the lowest point of the curve, and y^ is the length of the chord which is equal in weight to the tension at the lowest point. 284. A special solution of the preceding problem is given by the equations ^ = 0, lr = }^n, ^ cos 'Ij, The curve is the circumference of the circle formed hy the intersection of a horizontal plane ivith the surface of revolution. The tension of the — 144 — cJiord is the iveigU of a length of the same chord ivhich is equal to the dis- tance of theiolane of the curve from the vertex of the cone drawn, through the curve, tangent to the surface. 285. If 1/^ is the angle which the projection of/ upon the pLane of xz makes with the axis oi x, and if d\\)' is the elementary angle which two successive positions of y make with each other, this elementary angle and the radius of curvature are given by the equations -- — - d\\)' ^ m\l,d\\^ , \,^ — D^'yr + n.Y = sin^, A^/^ — A;^ = sin^,Z>,i// — cosl,I)yryf = mrpD.ii) — i>2,,sin^,. If, moreover, ic' is the length of the tangent drawn to the meridian curve at any point of the chord, and It, the projection of z/ upon the axis of^, the following equations are obtained, sin I, = u' . sin I, D, \\i = ii D, Y , i == ^' — i>^,sin;, = sin^,cos^,(- — i)^, log sin ^,^ ; which substituted in (143i9) gives, by dividing by sin ^, cos ^,, and transposing, i>„logsin'/= -J — . 286* In the case of the right cone, with the circular hasc, the sum of — 145 — y' and ii is constant ; if, then, a ^u -\-y, the curve is determined by the equation n 1 • . 1 1 1 I 1 r= — Z)„,log.sin.;,. The integral of this equation is sni "" ^*' («' - «' - yo) («' + y,^)-^ - («' + yo - 2 ^0'^ in which the constant is determined, so that iif may be equal to f/ when the chord is perpendicular to «'. The chord is also perpendicular to u', when and also when ^'' = ^ («' + ^o) ± i sj [{a + y',f + 4 «>a . When li is contained between a and ?/q, the expression for the sine of the angle which the chord makes with tt is less than unity, so that the angle is real. This angle is also real when u^ surpasses the greater of the roots of (1452i), or when it is algebraically inferior to the smaller of those roots ; but the angle is not real when tt' is included between these roots, but is exterior to the preceding limits It' and ?/q. The curve of the catenary itpon the vertical right cone consists, therefore, of three distinct portions, of ivhich one is finite, and included hetween two intermediate points, at tvhich the curve is perpendicular to the side 19 — 146 — of the cone ; tvhile the other iivo portions, commencing respectively at the two points, tvhich are the highest and loivcst of those at tvhich the curve is perpen- dicular to the side of the cone, extend to an infinite distance. These portions have two of the sides of the cone for their asymptotes, because the angle which s makes with ii vanishes, when u is infinite. 287. The finite portion of the catenary upon the vertical right cone may be investigated by adopting the notation «' + Vo — 2 ?/ sin^2r= 0082/":?, sin i a' — 7/f, sin^ = sin /sin cp ; and that of eUiptic integrals, of which the third form may be repre- sented by These equations give II z= \ {a -\-y'^{\ — sin z sec [^ sin 9) = 2K + ^o)(l — sec/isin^), , cos^j3 — sin^i sin ^(5 sm cos , cos d y/ (cos ^d — 2 sin 2(3) sin i cos d cos (p cos '^d — sin '■^ ^ cos'^0 — sin-^ p! ' tanl. 2/P sin-'/i' sin^cos^cos(Jp' D^ii =z — J (a' -\- y^) sin ?sec (i cos^ , V* COS'u/ COS /3 COS = 2 (^^' -h ^0 ) ( sec fi COS ^ — tan /? sin (S sec ^ ) , — 147 s =: Ka -\- ?/q) { sec (i ^^ (p — tan (-i sin (-i 7f\ (p ) ; r^ / tanl,Dou' tan.-i?\n^ tan p' sin, J JJ III = ■ — — — ' ' — 9^ u' (1 — sec p sin i^) cos 6* (1 — «sina)cos (1 — sec p sin f1) cos d (1 — n sin g ) cos Q tan j3 sin ^ sec'? ■ sin /tan -p sing sec /9 ?i^sin-qp for it is found, by differentiation, that i>,tan^-i^:^' = i>,tant-i^^^^^ •? tiinrjf 9 cos/^ sin i sin g (cos "d — sin ' / cos ^ g ) (cos '^d -\- sin ^ z cos - g) cos sin z" tan ^(3 sin (]p sec /9 1 — /I ^ sin •^ (p 288. The preceding value of the angle v^' admits of geometri- cal expression by means of the arc of the spherical ellipse in the form given by Booth. A spherical ellipse is the intersection of a cone of the second degree vAth a sphere of tvhich the centre is the vertex of the cone. Let a and l^) be the two principal semiangles of the cone, of which a is the greater, and w the angular distance of any point of the arc of the ellipse from its centre ; and its equation is obviously cot-oj 1= ^ — = ^ 7, {- - — -.- tan-oj tan''*^ ' tan-p Adopt the notation a =^ the arc of the spherical ellipse, — 148 — i = the angle which the perpendicular to either of the cir- cular sections of the cone makes with the axis, which perpendicular is called the cf/cUc axis, t = the angle which the focal of the cone makes with the axis, 1] = the angle of eccentricity of the elliptic base of the cone. If, then, through the centre (? (fig. 2) of the spherical ellipse, the axes AOA' and BOB' are drawn, and B joined to the foci F and F\ the sides and angle of the spherical triangle B OF, are BF=a, B0 = (j, OF=t, OBF=ij, BFO = ln — i, which are connected by the equations cos« = cos fi cost = cotijtanz, sin {i = sin a cos i =^ cot i; cot £ , sine = sin a sin 9] = tan z tan/:?, cos?] = cos / cos £ =: cot« tauf:?, sin ^ = sin i] cos (i =z cota tan t . Let C and C be the points at which the cyclic axes cut the surface of the sphere. Draw OP to any point of the ellipse, OF perpendicular to OF, C'// perpendicular to CF, 6^// perpendicular to Oil, i^'/r perpendicular to OH; take FK equal to 00, and, draw Z 7)/ perpendicular to OA. If, then, ^=zL3f, cp=:LF3I, l^IIOC, V^OCE, the following equations are readily obtained, C0S2 = cos 00z=. cot /I' tan ^, tan^ := cosztan?/ = cos^/tan^ = cos " /cos fc tan (/J = cos/cos)jtan(/) , — 149 — sin ^ = sin z'sin cp , SQG^'^ z= 1 -[- cos^/cos^)^ tan^9) = sec^cp {cos^if) -\- cos-^cos^/jsin-cj)) = sec^c/)(l — sm^i]sm^(p -\- sin^7^sin-t^) = 860^9 (cos^t5 — sin^j;cos^^sin^y), 2 o 9,>/ii 9'i9\ l-l- COS - i tan - qp cos^'oj = cos-«cos-"(l + cos-^tan-cp) = . ^ ^ :r.—^^ , 9 cos^ « ( 1 + cos^ ^ tan^ rr ) cos ^ a cos ^ 9 1 -)- cos "^ j^ tan -^ g) 1 — siu - // sin - go ' . 9 sin-«cos^(rsec^^ sm-oi = ■ 2 . 2 "^? 1 — sm '' // sin '^ go ^ ^ ^ cos j; cos r cos ^ ^ cos^^cos^sin2« 'l"^ cos-g) sin''^o)(l — sin-/^sin -qr) ' •r-w cos ^ « sin ^ /? sin ^ /5 sin nr cos ff' x/ (iJ 1 -. r^ — '? (1 — sin ^;; sin ^ g) ^ sin ft) cos to' j^ 2 sin^^cos^^j/sin^j^sin^'/sin^qccos^" ^ cos - (9 ( 1 — sin - 1] sin " g) ^ ' • 9 T-i ,,9 sin- J cos-/; cos-" ■ "■ cos'g(l — sin-z/sm-qp)^ i>,a2^i>^cu2 + sin2to7>^," sin 'j3 cos ^7/ cos^(?(l — sin^j^sinqc)^ V cos^qcsec-^^ sin ^ |3 cos ^ ?/ /cos - — sin ^ ?/ cos - ^ sin - g \ cos-0(l — sin-?;sin-g)- \ cos-gosec-^ / sin -j3 cos ^7/ /cos-/?sin-/^-sin^(]f cos^/? -)- sin-z/sin-zyN -) sin -p cos-/; cos-6 (1 — sin^?;sin^g) -' D^a = sin |j cos j; sec 1 — sin^z/sin^qo' o = sm[-icosi]^,{— sin-?;, c/)) = ^''^||^'^' ^,(— sm''i, f/') 289. In the particular case in which = iJt, this equation is, by (I4817), reduced to o — tan/^sin /:? 3'V(_ 7?^, (j;) — 150 — which substituted in (I475) gives, VI ^= o 4- ian^ ^^——. I ' tan cp 290. For the length of the arc of the chord which extends from its lowest to its highest point, this equation becomes and if the magnitude of this angle is commensurate with the total developed angle of the cone, Uic chord returns into itself, after imssing around the cone once, twice, or several times, dependent upon the magnitude of the angle of the cone. 291. To investigate the infinite portions of the chord, let /o and /i be the roots of the equation (1452i), and the equation gives 44 = — «^o- Adopt also the notation of § 287 and ^ sin i (a' + y^ — 2 u') cos p' (a'^f, — 2 u') — r,-\-r, — 2u" sin^'=:: sin? sine/)', and the following reductions are obtained, by the substitution of cosec^' for sin (5, ?f'= l(lo-{-l[){l — secficosecg)'), /> ,s — 1 (r/ -J- 9/\ {^_^IlM^ sec^cosVX '^'■^ — 2 l^* -1- !/o} {-^^r sl^T^^^ j = K^'+yo)[tan/'isin/'lseci3' + sec/'icos(3'+ sec/':?Z)^,(cos^'cotc/)] , «== K«'+^o)(sec fill S.f/+ tan fi sin /ig^,f/+ sec /icos^' cot 9')> — 151 — —^ , sin ^ p cos /? sin - g;' sec 0' — sin - ^ sin g' sec d' ^/ ^^ ~ 1 — cos-/3sin2,y/ tan /? sin /j sec 0' , ,, . ,., ../ i t\ x. =z — -^ — .!\ . ., , — tan :? sin j sec ^ + -^n' tan 1 — C0B-(Ssm-qj ' ' '9 [_1]CO50' cos q' ' 1// = tan/isin/i 5?,(— cos^.^i^') — tan.isin/'i^ic/ + tan^-ii ^'. 292. The term of the prececlmg value of i/'', which depends upon elUptic integrals of the third order, may be constructed by means of a spherical ellipse, of which the parameter is the reciprocal of that employed in the construction of the similar term of the finite portion of the chord. The parameter of the spherical ellipse of § 287 being sin?], the reciprocal parameter is sin i ,j := COS p , sni;; and the length of the arc of the corresponding spherical ellipse for the amplitude 9' is n'= sin/icos,)g>,(- cos^i, 9') = ^i^^ 3>,(_ cos^'i, ,.'). This arc is reduced, in the case of to o' = tan (3 sin tj^.{— cos^ ('j, (p') . 293. The finite portion is exactly circular when d — ij^. In this case 2 = , [/^ ^ J TT = « , and the equations of the infinite portion become ?«'= d{\ — y/2.coseca)'), s = rt y/ 2 cot 9)' i// = V^2(i-7t — y') — 2tanf-ii[(v/2 — l)tan(i7r— 1^/)]. — 152 — 294. As ^0 diminishes from the vakie a, the finite portion becomes more and more eccentric, until when hoth the finite and the infinite iwrtions degenerate into straight lines, ivhich are the sides of the cone. 295. When g a is negative, a and g^ cease to he the limits of the finite portion, and lecome the limits of the infinite portion, ivhile ^ and l[ become the limits of the finite portion. But Iq and l[ are imaginarg, if g^ is included bettveen the values /o=:(-3 + 2v/2)a', so that between these limits the finite portion disappears, and the cJiord con- sists onlg of the two infinite portions ; and at the limits the finite portion is circular. To investigate the infinite portions between the limits, in ivhich the finite p)ortion disap2)cars, let tan i' =^ sin i^ — 1 , sin ^" =1 sin i sin cp" ; and the following equations are obtained by simple transformations, sin(:Jcosr= y/|, u'=z 1 («'_|_^^)(1 — sec fi sec 9") = |-(a — ^o)(cos/i — secy"), cos^(3'= 1 -|- tan^«'cos^9" = sec ^ (i — sni''^ sm 9) ) =: sec'^i cos^d'^; ^^''*— 2V« -h^oiV cos>"cos0" sJ'l.co^O")' 1/ / I /'x/.'=ec/? .,/ '„ sec/?c' t/ tan/?(-jj ,A — 153 — ^ , V^ 1 tan /? sec ^' i /i i. o \" n +^ . l-ii *=^"''' D „w =^ — T-^^^ — ^. — ^-T/ + V o tan li sec (3 — D „ tan ^ '^ — -;, 9 ' 1 — cos-p^cos-g-' I V- ( V tune cos ^' cos /? sec 0" ■ ./ ^^ ,v/ ^= \ ., + COS i COS /5 sec 6 1 — ^ sin " (jp ' ' - i?,„tau.->.(cos/^cos^"^) - i>,.tan-«,^;, i(/ = — coai'coi'ii 9, (— 1 , (j") + cosj'cosj'i Sfj^^" ^ _ cos^"cos/5^,,(— 1 , ^'') + COS^'^COSflJ^^i,/ r_n tan -t' 4- COS/? COS g/^ tan -(9^^ ^ ^''^^^ tan r tan 6"( 1 — cos /? cos g") ' in Avliich the elliptic integral of the third form admits of interpreta- tion by means of the arc of the spherical ellipse. 296. When the negative of ^o is equal to a the equations may be greatly simplified and reduced to the following forms, COS/? =: 0, cosr= y/o-, ^9"^' — ^(2 — sin--'g.")' , sin g cos^^/ =1 — 2 sin 2/ = cos2i//. / 9 U cos - g-" cos 2 Ti;' ' 2fAM 25 //^^ ;w/«r equation of the equilateral hi/perhola. In this case, there- fore, the curve of the chord upon the develojjed cone is an equilateral hjper- hola ; this case was recognized by Bobillier in an imperfect investi- gation of the catenary upon the surface of the vertical cone of revo- lution. 20 — ]M — 297. When f/ic surface of revolidion is an ellipsoid, of which the equation of the vertical section made by the plane ?jx is, let a sphere be constructed upon the axis of revolution as a diame- ter, and let g) be the angle from the vertical point of the sphere to a point of which ?/ is the ordinate, so that y ^=^ A cosf/) , X = Bsin (p , u =z A{seG(p — cosg-)) = yl sin c/) tan g) , I)^?/ = — Asin(p. These equations, substituted in (14499), with proper regard to the different position of the origin of coordinates, give i>^logsin^,, = — ^sin9)i:>,logsin^, = —coUp + ^J^^^j^p sm v^ ^' sin (f (cos cp -j- 31) -| sin 2 g) -|- Msinq) in which iVand 3/ are arbitrary constants. 298. The maximum and minimum of sin ^/ are determined by the roots of the equation cos 2 9) -|- 3Icos(p =z 0. If these roots are (p' and y", the equation gives cose// cos 9)" -]- 2= 0? 31= — 2 (cos 9' + cos 9") = — 4 cos 1 ((// + (p") cos } (c// — (p') = sec 9' -|- sec cp". Of the two roots, therefore, one is obtuse, while the other is — 155 — acute ; if one is contained between ^ n and fTt, the other is impos- sible ; and when both are real, one is confined between I n and | n , while the other is without these limits. The corresponding mini- mum and maximum values of sin I, are N T N -, and tan(jp'sin^(jp' tan qt" sin -^ gj" * Both these, independently of their signs, are minimum values, and when they are both absolutely greater than unity there is no catenary ; but if either is less than unity, there is a corresponding portion of the catenary. AVhen both values are less than unity, the catenary consists of two separate portions, because there is between ^/ and (^" a value ^'" of ^ which satisfies the equation cos (/J = — M, and the values of / ,// cos-ff/ — co.s2f/ sin-n[/ COSO) COSCp = *- ^, =^ r, ' ' cos(j& cosg, • 2 " cosf/) — C0S9 = ^^-V = ism^9 cosf/' , are positive. 299. The csj^ecial case of gives 9^^ = 171, 31 =z ; sm; ' ^ sin 2 (]p •* and each of the minimum values of sin ^, is 2N, — 15G — which, behig less than unity, may be expressed by 2i\^=:sin2«. This equation gives . , sin 2 a ■' sin z fjf If, then, \ is determined by the condition o , cos 2 (B C0S2/w ^ r,-, COS z « simple reductions give 5 y/(cos^2« — cos'^2fjf) cos 2 « sin 2 P. COS^/^::^ -. -^ : ; , y Sin 2 (jp sin 2 r; ' +.,,. _tan2« sin 2 ;. ' 2 « si sin 2 g) T-v cos 2 « sin 2 P. A/i^ = .;n9^ == COS,,, 1^ = y/(sin> + -pcos^f/.^sec;,, = v/[Kl+?) + i(?-l)^««2«cos2x]; -p. D^s?k\^\. sin2«Z)fflS ^ ' i3fsin 9 ^siu ^ sin 2 qp ' 7-v sin 2 « Z)- * "- ' xjsin q sin2 (p In the case of the iwolate ellipsoid, the notation . o. 2(i?2_^2)cos2« sin"? zzzi - - i?^ + ^^+(i^^ — ^l-^)cos2, siu^ =r sin?' sin P., gives the equation B =z v/(i?2cos2« + yl^sin^ft) $. I . — 157 — In the case of the oblate elUjJSoid, the notation >/ = i7i — ;., . ov 2 (.4-^ — ^2) COS 2 « sint5'= sinrsin^.', gives the equation s=: y'(^2cos^« -{- B'^^m'^a)%^l'. In the case of the sphere the equations become s = Al; and this result of this case is obtained by Bobillier. This case also gives the equation 7-x sin 2 a '- ' sin q, sin 2 qp sin 2 « (sin^a-|- cos 2 a sin ^ A) y/ (cos ^ a — cos 2 cc sin "^ A)' which by the notation cosr'= tana, sin 6'' = sin i'^ sin X , becomes ^ 2sec^^^ ^'^^ ~~ sin«(l+tan2i"sin2;.) ' ' since ^ ' ■' = 2sin«tan^a^i//( — sec-«sin^r, I) -|- 2sin«9^,y,A , ^, r i,sinj"tan(9"cos?. -[--tan^~^^ 5111 « oOO. Eeturning to the general case of the ellipsoid, let — 158 — a and /? be the limiting values of 9 for the upper portion of the curve, and a and /i' the limiting values for the lower portion ; and let Hence the following values of M and N are obtained N=^ \ sin 2 a -|- 3/sin a == | sin 2 /:? -f- 3/sin [^ , — N^\ sin 2 a' + J/sin a' = J- sin 2 /f + J/sin /r, T. ^ COS f COS 2 // cos t' cos 2 // cos // cos // ' iV^= tan?^ (COS"-?^ C0S2€ COS^i;COs27y) = |tan^(cos2£ — cos27j) = tani^(cos^£ — cos^j;) = |-tani/(cos2i/ — cos2fc')^ tan?/(cos^'}/ — cos^t') =1 tan -)] sin « sin /5 = — tan r[ sin a sin {^' . 5 sin ?/ sin « sin (3 — sin?/sin r/sin^' ^ sin (f (cos 1] cos gp — cos e cos 2 r^ sin qp (cos // cos (jp — cos e' cos 2 //) ^ Y'[ — (cos qp — cos «) (cos fjp — cos ,5) (cos q? — cos ic') (cos qp — cos^')] ' ^ sin g:(cosqp -j- J/) vT — (cos^qp — 2cos?/ cos£ cosqp -(- cos« cos|3) (cos-qp — 2cos?/cos£'cosqp -\- cosa'cos^')] sin qp (cos qpj -|- M) y/ [sin ^ y (cos q:. -f J/) ^ — jyrg] sin qri (cos qp -|- M) The numerators of the first and last values of cos^, give, by direct comparison, — 2 Jl/= cos« -(- cos/i -|- cosa'-|- cos/f = 2 cos?j cos£ -|- ^ cos?/cose', whence cos 1] cos 1/ cos t' = ( COS 2 ij — cos ^ jy ) COS £ =: — siu ^ 1] cos e , cos 1^ COS 7/ cos £ = — sin^jy'cost', — 150 — cos^j^cos^r/ — sin^T^sin->/= cos(»; +^/')cos('/ — i^) = 0, The comparison of the values of jV, shows that the value of )/ must be obtuse, whence cosfc' = tan>^ cose, cos£ = — tan?/ cos c'. 301. The general case of the surface of revolution admits of one integration, by denoting by v the ordinate of the meridian curve of revolution, which gives this equation, substituted in (14429), gives, by integration, sni ,,/ — , - , in which Vq is the ordinate of the meridian curve at the origin. This form of the equation is, however, limited to the case in which the curve has a point, in which its direction is horizontal. But every case is included in the form M sm ^' ^'(y + yo)' in which J/ is an arbitrary constant. 302. In the case of the surf ace formed ly the revolution of the equi- lateral hyperlola about its asymptote, which may be called the equilateral asymptotic hyperholoid, if the equation of the revolving hyperbola is — 160 — the equation (159v4) becomes %n\y, — — , and, therefore, the inclination of the curve of this catenary to the arc of the meridian is constant. When J/ is greater than 1)"", the curve is impossible, but when the catenary becomes a horizontal circle, and 303. It may be inferred from the comparison of the two pre- ceding sections, that, upon the circle of intersection of any surface of revo- lution ivlth the equilateral asymptotic hyperholoid of equation (ISDsi), the arc of the catenary of cither surface makes the same angle ivith the meridian curve of that surface. Hence, the limiting hoiizontal planes of the catenary of equation (ISOig) are the intersections of the surface of revolution upon zvhich it lies ivith the equilateral asymptotic hyperholoid, of ivhich the equa- tion is The catenary extends over that portion of surface ivhich lies exterior to the asymptotic hypcrloloid, and does not extend over that portion of surface ivhich is included ivithin the hyperholoid. 304. To complete the solution of the catenary upon the equilateral asymptotic hyperholoid, the equation (ISOgi) gives tan^, = — D,,v^= J—, — -2, whence the following equations are obtained ; (^+.yo)'=^'cot^,, — 161 ^"yy = ~ 2(y + yo)sin^^^ But it is found by § 285 that Tj secf^rtanlr (y -\- 1/,) tan;, whence B- w z= ^-^ • >J' t 9c5ti2.'/ ^rtc.v ' 2sin^'^,cos^. of which the integral is o^ 305. If the chord is not strictly confined to the surface so as to be incapable of removal from it, but if it simply lies upon the surface, without the power of penetrating it, it must leave the sur- face whenever the pressure becomes negative, that is, when the sign of jR, computed by (14229), is reversed. The points at which the chord leaves the surface are, therefore, determined by the equation 21 — 1G2 — CHAPTER IX. ACTION OF MOVING BODIES. CHARACTERISTIC FUNCTION. 306. Related to the idea of the potential, and, in some respects including it, is that of the action of a system as proposed by Maupertuis. Every moving body may be regarded as constantly expending an amount of action, equivalent to the power which its motion represents, that is, to the product of the force of the moving body multijDlied by the space through which the body moves. Hence, with the notation of Chapters H. and IH., if F designates the whole action expended by the system, the action expended at each instant is and the total expenditure of action is The function V is called by Hamilton the characteristic function of the moving system, and he has resolved the problem of dynamics into the investigation of its form and properties. 307. If the power, with which a system is moving at any instant, is denoted by T, its expression becomes, by {^20), The preceding expressions for the expended action give, there- fore. — 163 — V==f2T. rEIXCIPLE OF LIVING FORCES, OR LAW OF POWER. 308. If 12 denotes that function which, in the case of the fixed forces of nature, is the potential of the moving system, its change for any instant is, by (3424) ^''^^ § ^8? Hence, in the case of the fixed forces of nature, if U is an arbi- trary constant, which is only the analytical form of the proposition of § 58, and is caWedi tJie principle of living forces. The term living force denotes the power of a system, so that this principle may, with equal propriety, be called the laiv of power. CANONICAL FORMS OF THE DIFFERENTIAL EQUATIONS OF MOTION. 309. The equation (815) may be written in the form = D,:^i{ni-^v-^dS]) — ^^{inii\d DtSi) If, then, r^^,r^2,'^hi etc., are assumed to be the independent elements of position of the n bodies of the moving system, 81,80, etc., may be regarded as expressed in terms of these elements, so that v=.D,8 = 2:^{D^8D,i]). — 164 — With the notation this equation is resolved into the equations represented by The substitution of these values give, if T^j^^, denotes T ex- pressed by means of i^i, i]^, r(^, rf^, etc., whence 2),i2 = (i>,Z>„,-i>,)r,,,,. This expression represents the elegant forms of the differential equa- tions of motion given hg Lagrange ; but the mode of investigation is adopted from Hamilton. 310. In the special case, in which the independent elements of position are the rectangular coordinates, x, g, z, of the different points of the system, these equations become D^, T^^ 2-, = mx' = mDtX, Z^a- i2 = m Dt X == m D'\ x . When the coordinates of the system are subject to conditions, these equations are still applicable, provided that the forces, by which the conditions are maintained, are included in the forces of /2, or more properly of d £1. The values of D^Q^ and D^il can be obtained from the given differential expression of 12, even when — 165 — such expression is incapable of integration ; for this form gives 311. By means of the notation ''/I? '^i2j •••• etc., may be eliminated from the value of T, and Tjj^^ may denote the resulting value, expressed by means of ?;i, 7^2, Wi, W2? etc. Since T is a homogeneous function of two dimensions in respect to i]\, i]\, etc., it satisfies the equation whence But the variation of T, derived by the usual method, is which, subtracted from the previous value oi2d T, leaves This equation is equivalent to the two equations T) T — B T and Lagrange's canonical form assumes the folloiving expression given hy Hamilton, 312. But /2 is, in the case of the fixed forces of nature, a function of i^i, 7j2, etc., without other variables. If, then, in this case. — 166 — tlie preceding equations assume the simple form which are given by H^vmilton, in which X2 may involve the time. VARIATIONS OF THE CIIAEACTERISTIC FUNCTION. 313. The variation of the characteristic function, taken npon the hypothesis that the time does not vary, is But, from the preceding equations, the sum of which and of the equation is The variation of the characteristic function is, therefore, ^F= ^jj{codi] — cOqSijq) -\- id II, in wdiich coq and ijo ^I'e the initial values of to and ij . If, then, V is expressed as a function of the initial and final coordinates, ?;, tu, i^q? — 167 — and ojo? ^^^^ of the constant II, its derivatives are By means of these equations, the prollcm is resolved hj Hamilton into the determination of the single function V. 314. In the case in which the independent elements of posi- tion are the rectangular coordinates, these equations become 10 = m X = m Dt x == D^ V, ioq = w?.ro = niDiXQ = — Dx^ V. 315. If the expression of the forces involves the velocities the final expression of dT in § 313 is incomplete, and the present mode of investigation is not easily and simply applicable to such cases, which is of less importance, because these cases are not, in the most comprehensive view of the subject, the cases of nature. PRIXCIPLE OF LEAST ACTION. 316. When, in the case of the fixed forces of nature, the ini- tial and final positions of the system are given as well as the initial power with which the system is moving, the variation of the charac- teristic function vanishes, and, therefore, the function is generally a maximum or a minimum. The action expended by the system, which is measured by this fimction, is also a maximum or a mini- mum ; or, in other words, the course by which the system is com- pelled to move from its initial to its final position through the action of the dynamic laws, is that in which the total expenditure of action is a maximum or a minimum. But it is obvious that, in most cases, and always, when the paths in which the various bodies — 168 — move are quite short, the described course cannot correspond to the maximum of expended action ; and, therefore, in most cases the sys- tem moves from Us given initial to its given final position ivith the least possi- ble expenditure of action. Many examples can, however, be given, in wliich the expended action is, in some of its elements, a maximum ; although, even in these cases, the expenditure is a minimum at each instant, or for any sufficiently short portions of the paths of the bodies. 317. This principle of least action was first deduced by Mauper- TUis, through an a priori argument from the general attributes of Deity, which he thought to demand the utmost economy in the use of the powers of nature, and to permit no needless expenditure or any waste of action. This grand proposition, which was announced by its illustrious author, with the seriousness and reverence of a true philosopher, is the more remarkable that, derived from purely metaphysical doctrines, and taken in combination with the law of power which likewise reposes directly upon a metaphysical basis, it leads, at once, to the usual form of the dynamical equations. 318, To deduce the dynamical equations from the combina- tion of the principles of least action and living forces, add together the two variations of T., If the sum is introduced into the variation of V, the result, reduced by the condition that at the limits of integration, becomes — 169 — The factor of ^t;, in this expression, must vanish by the princi- ples of the method of variations, which gives immediately the gen- eral expression of Lagrange's canonical forms. PRINCIPAL FUNCTION AND OTHER SIMILAR FUNCTIONS. 319. The function jS determined by the equation is called by Hamilton the principal fimdion, and its variation deduced from that of Fis, obviously, dS=dV—idH—IIdt = .2',(w^7^ — Wo^7]o) — Hdt. Hence, if S is regarded as a function of t^, t^q? '^j '^o? ^tc, with the time t, its derivatives are The principal function may, therefore, be used in the same way with the characteristic function in the determination of the motion of the system. 320. Many other functions, as suggested by Hamilton, can be substituted for the principal and characteristic functions. Thus the function gives 22 170 — Hence, 321. The introduction of Q = W+ tll=f(^,{ri o/) + //), gives, in like manner, ^u>Q = v> ^"oQ = — Vo} 322. Other functions can be formed by the combination of V and W, or 8 and Q. The combination may be such that for some of the coordinates, the function shall have the same form as V or >S', while for the remaining coordinates it shall have the form of IT or Q, and the function or can be substituted for V or 8. 171 — PARTIAL DIFFERENTIAL EQUATIONS FOR THE DETERMINATION OF THE CHARACTER- ISTIC, PRINCIPAL, AND OTHER FUNCTIONS OF THE SAME CLASP 323. By substituting in the equation for 1], to, etc., as well as for t and II, their equivalent expressions, as partial derivatives of V, S, W, Q, U, and P, partial differential equa- tions are obtained, the integrals of which give the values of these functions. To facilitate the expression of this substitution, T and 12 may be assumed to have such functional significations that The partial differential equations are, then, T{D^ w,m) = n (- n^w,D^ w) + II, 324. When the independent elements of position are the rec- tangular coordinates of the bodies, these equations become, by the notation of (543i), ^n.{l nv) = 2n{D^v,x) + 211, :^,M^" +!/" + ^") = 2f2(- D,,W, Id^.w) + 2IL — 172 — S^mix'' + S/" + n = ^^{t,^.^.'Q) + ^^.Q- 325. Through the preceding investigations, the forms are developed by which every dynamical problem can be expressed in differential equations. It only remains, therefore, before applying these forms to especial problems, to consider those methods of inte- gration which are best adapted to their discussion. CHAPTER X. INTEGRATION OF THE DIFFERENTIAL EQUATIONS OF MOTION. 326. In discussing the differential equations of motion, it might be permitted to suppose a previous knowledge of all that has been written upon the integral calculus. But since the profound philosophical views, with which this subject has been illuminated by Jacobi, have not yet passed from the original memoirs into the text- books, a development of them is required by the plan of the present work to facilitate its further progress. I. DETERMINANTS AND FUNCTIONAL DETERMINANTS. 327. If {n-{-iy different quantities are given, which are represented by — 173 — in which every number from to n can be substituted for k or for the number of accents denoted by i ; and if all possible products of {71 -}- 1) factors are formed similar to in each of which the same number is never repeated, either for Jc or for i ; and if these products are successively formed by mutually interchanging two of the inferior numbers, and at the same time reversing the sign of the product ; the sum of the products has been called by Gauss the determinant of the given quantities, and may he repre- sented hj ^^ = :E±ad^di a^:\ Thus, for example, •3^0 = -^i ^^= ^? ^1 = ^ + adx = aa\ — a^d, 9^2 ^^ -^ i ^^1^2' = ad^a'l — ad--!) ==0, the form of the determinant may be simplified. For it is evident from inspection of the fundamental product, regarded as separated into two factors, that every elementary prod- uct, produced by an interchange between the inferior numbers, such as to transfer one of these numbers into the second factor, vanishes, and may be neglected. Hence ^,=:E±aaw; «ir-i'^:^±«ir«tr4^i'^ a':' if 331. When, in the preceding proposition, m is equal to n, so that it becomes for, in this case, — 175 — 332. When, in addition to the preceding equation, the values of the elements vanish, for which m is equal to n — 1, so that f,(i>n-\) (-) "A, is ^„4 = ^''^^kU + ^W + ^t'u'"' = ^ii^M'u'"') 23 — 178 — 340. If all the quantities ii, u', n", etc., vanish, i, t^, t2, etc., must likewise vanish, unless the determinant vanishes.^ If, therefore, either of the quantities /, /i, ^25 etc., does not vanish, when it, u', ii', etc., vanish, the determinant must also vanish, whence the equation ( 17613) applies even when i' = /, or for all values of z' Hence, it is evident that t-.t^-.t^ : /„ = oM) : 04'/) : ok^ : (M^. 341. The process, by which the value of /„ was obtained, may be regarded as designed to eliminate the n quantities t,ii,t2 4_i from the given equations. By precisely a similar process, the m quantities t, t^. t^ tm-\ ^^^^ be eliminated from the first m -|- 1 of the given equations, and the form of the resulting equa- tion must be Bu + B'i{ + + i?"")w<-' = CJ,, + C'^.+i^.+i + -VCJn, in which In the same way, if r^ = :E±Gk'A\oA'i Q^lf), the quantities «''" + ^', 2<<'" + 2', ?(<"' may be eliminated from those of the equations (1773o) which give the values of 4o ^2 + 1? 4? — 179 — and tlie form of the resulting equation is Eu + E'u' + + E^-hi^-^ ^ FJ„, + i^,.+iC + i + + FJ,,, in which But the two equations, obtained by these processes, must be identical in the ratios of their coefficients. Hence or OV. m iJV. n' m-ri, It '->i ' »i + 1 j /i or by extending the series of ratios to all values of m, But it is easily seen that 'n, H "^ n ^'-/i — 1, and whence ^1'" Qjk'^'"' >\. = ^^r, = a1Rl-K A repetition of the same process, in a different order, upon the given equations gives Hence — 180 — 342. The ratio of the values of r„ and ri^,^ may be prefixed to the series of ratios of (ITOig) in the form The series of ratios gives, then, ^^ » • ^^m ' « • ^7n I m + 1, h J or This investigation is derived from Jacobi. 343. The variation of a function of the quantities represented by «i.'' is expressed by the formula If, then, the values of the quantities, denoted by «*'', are such that and if the corresponding values of t,t^, ..... . /„ are denoted by ^*^', /]", t\^\ the expression of /^^' assumes the form and therefore = (^^„+^,.,(q4(:)(z;/,)). 344. If the given quantities are such that — 181 — it is readily perceived that and whicli is given by Jacobi. 345. A system of equations, similar to those of §339, repre- sented by the form gives, in the same way, If an equation similar to (ISOge) is derived, 346. Let the {n-\-lY quantities, represented by c^l\ be derived from the given elements a^l^ and Z»i!* by the formula and let the determinant of these quantities be -a^„=^±.e;4' c\:\ If only one term is taken in each of the quantities ciJ', the general term of 3„ is represented by A-fAm) .(m').Amf') ]Am)Um')Umf') ^(li ilj/ a i// U 1^ i^f O i^f — 182 — A mutual interchange of the letters A", followed by a mutual mterchange of the letters i in the resulting terms, j^rocluces all the terms of ^„, which correspond to the same combination {31) of accents m, m^ , etc. A different combination of accents gives a dif- ferent set of terms ; and if w^'"^ =^^± «""'«f ''«r'"» «j::f ^ q^(M) ^ ^ 4_ j(«)^(-.')^(«/o li^'p^ denote the determinants of the given elements corresponding to one of these combinations, the complete determinant is expressed by =-^» ^ ill V '^a ^ n )} which is given by Jacobi. 347. In the case of 27 = 11, there is only one combination (31) of the accents, so that in this case n n n 7 which was given by Cauchy. When there is no combination (31), in which all the accents are different from each other, and, therefore, it follows from § 329 that, in this case ^ 0, and that, in all cases, the combination (31) must consist of accents which differ from each other. 348. In the special case of «i:' = ^i;', — 183 — which gives cl!)=c|^\ the value of the determinant is reduced to which, 'when is reduced to 2^ = n FUXCTIOXAL DETERMINANTS. 349. If the given elements a^l^ are the derivatives o^ (n -\- 1) functions f,fi, /^ of (;i -]- 1) variables x,Xi, :r,j, so that the determinant of the elements is called ihc functional deierminant of the given functions. Thus, in the present case, all the terms of the determinant are obtained either by a mutual interchange of the variables, or by a mutual interchange of the functions, the interchange being accompanied in either case with a reversal of the sign, precisely as in deducing the terms of the ordinary determinant. The proposi- tions, which have already been given in reference to determinants, are easily applied to functional determinants. 350. In the case in which all the functions, above the {in -j- l)st, are free from the first m variables, the condition of (1749) is satisfied, so that the notation of (17422) gives the equation (1742o) n 771 — i ?W , 71 ' — 184 — 351. In the case in Avliich every function is free from the variables of which the inferior number is less than that of the function itself, the equation (175io) is satisfied, and the functional determinant, reduced to a single term, is %. = DJD^J,D^J,^ A„A. 352. If the given functions are not independent of each other, the determinant vanishes. For if the equation, which denotes their mutual dependence, is expressed by 77=0, its derivatives, with regard to the given variables, are represented by the equation The equations, included in this form, are identical with the linear equations of § 339 when the values o^ii vanish and All these values of ^ cannot vanish, because the equation, which expresses the mutual dependence of the functions, must involve one or more of them ; and, therefore, the determinant must vanish by § 340. 353. If either of the given functions (f) contains any of the other functions, these functions mag J)e regarded as constant in finding the fmictional determinant. For each derivative of /^ is the sum of two parts, one of which is derived by direct differentiation w^ith refer- ence to the variable explicitly contained in the function, and the other part is obtained by indirect differentiation through the functions involved in /. The Avhole determinant may then be regarded as composed of two such portions. But the portion of the determinant obtained by the indirect differentiation of/^ is the — 185 — same as if/,, not containing explicitly any variables, were .simply a function of the other functions. This portion must, therefore, vanish, and the remaining portion of the determinant is that which is obtained by direct differentiation, conducted as if the functions, involved in /, were constant. This proposition is applicable even where several of the given functions contain the remaining functions; but not when they mutually involve each other. 354. If the second of the given functions contains the first, if the third contains the first and second functions, and if, in general, each function contains all the previous functions, the preceding proposition is applicable. Hence if, b}^ means of the first function, the first variable is eliminated from all the other functions ; if, by means of the second function thus reduced, the second variable is eliminated from all the subsequent functions ; and if this process is continued until each function is liberated from all the variables designated by an inferior number, although it may involve all the preceding functions ; the determinant is reduced to a single term as in § 351. This will often afford a convenient method of obtaining the functional determinant. 355. In performing the successive eliminations, the operation must not be restricted to any prescribed order of the variables, but one of the variables, remaining in/, must occupy the place of a:,. Hence there is not one of the factors of the determinant in the form of § 351 which vanishes, unless a function be obtained from which all the variables are explicitly eliminated, or, in other words, unless one of the given functions is included in the others and can be derived from them, so that they are not independent of each other. If, therefore, the given functions are mnlmlly independent, their functional determinant does not vanish. 356. li F,F^, F,, are given functions oi ff, fy, 24 — 186 — which are themselves functions of the variables a;, Xi, x„, the derivatives of the functions (i^) with respect to the variables (x;) are represented by the equation This equation coincides with (I8I24), if the notation for «^' is combined with the notation The remaining notation and conclusions of §§346 and 347 may, therefore, be a^Dplied to this case. Hence, by (182i8) ilie functional determinant of the independent functions {F^, taken tvith respect to the same number of varialiles [x,), loldcli enter into {F^ only as they are involved in the same number of independent functions (f) explicitly involved in (Fi), is obtained by nmltiplying the functional determinant of (i^) taken tvith respect to (f) by the functional determinant of (f) taken tvith respect to {Xi). If the number (p -\-l) of functions (f) exceeds the number {n -\- 1) of functions {Fj), the complete functional determinant of [Fj) is by (182n) the sum of all the partial determinants cf (Ff) obtained by every possible combination of {n-\-\) of the functions (f). If the number of functions {f) is less than that of the functions (Fi), the fimctional determinant vanishes, as in (I8225), tvhich corresponds to the proposition that the number of independent functions cannot exceed the num- ber of variables, by tvhich they may be expressed. 357. In the case, in which F — X- all the derivatives of {F^) with reference to the variables {Xi) vanish, — 187 — except those included in the form In this case, therefore, is the functional determinant of (a-,) regarded as functions of (/,), and the equation (182i8) becomes or the functional determinant of (rr,) taken vntli respect to (/,) is the recipro- cal of the functional determinant of [fi) taken toith respect to [xi). 358. If in the linear equations of § 339, the values of {t) are expressed by the formula either of the equations is represented by unless m = i, in which case This value substituted in (ITTs)) gives 359. If it is again assumed that 188 — the equations of § 345 give ^,//^Z>^/; D.r F Df ]? Df F = - + D^F'l).r,Fl Dx^Fl ^ ± ^/ ,:i^. +i^/„..i^.+2 ^f,K. 2±DyFJ)f^F, ^/^„ _ -^ 4- n Fi Df F] Df F^^A- Df F ^^Df F ^o Df F But the equation (ISOge) is applicable to this case if (i^) is changed to {F}), and, therefore, the introduction of a common factor into the terms of (1892g) gives, by means of the preceding equations, 6S _ ( y, + 1 ^±D^FD^, F, D^,,F^ Df„+, F„^, »— V >* ^±D,FD/iF, Dr^F^, o 65. There are various interestino; and instructive relations o between the partial determinants of functions which have been developed by Jacobi, and which will be found useful in discussing the theory of differential equations. If the number of the functions (/j) as well as of the variables (.r,) is increased to ni -\- n -|- 1 , let mi) — ^4-7) fDx f, Dc f .Dx / , . If, then, from the function (Z,,^,), all the variables x, x-^, rr„_i are eliminated, and the functions /, /i fn-i introduced in their places, and the function {fn+i) thus transformed is denoted by {f\j^i), the values of 0^ become 2B'') = gic ,Dx /-i... k n — 1 n->rkJ n-\-i • The determinant of the (m -{-If functions (^i^') is, con- sequently^, ^ + ®5B;2B;' S^c")— .'7|5'«+i v-i-z>;r f^Dx /•! -, . D.r /I . -!- 1 ^ in JVun — 1 — _L nJ « n + lJ « + l « + m./ n-\-m' — 191 — But it is obvious that whence ^ + ^% ^\:p = vji:r-i ^'. + m ' 366. If '^'k^ denotes the value which 3^'^„_i assumes when all the derivatives relatively to Xi are changed into the derivatives rela- tively to x,,^;,, it is evidently the factor of ^-e^fn^i in the value of — ^J;!'. In the value, therefore, of the determinant ^ + 2B'5'); SB'-), the factor of D,f„^-rJ], + i ^■'■nfn + m is ^y-+-':s±%%[ ^^1^^ But the factor of the same quantity in ^^i+m is? by inspection, ( \m+i-^\j)^ fD^ f Br f Dx /• , , Dx f , \ ) ^ _L nJ n + lJ I n + mJm ,„ + i./w! + l « i J n —\ It, therefore, follows from (lOl^) that ^ + SB S^i '5]',;;" \ / w + l'^ J_ m + iy m + 'lJ I m+nJn — \' 367. The factor of ^.^._J„ + , in the value of 5]';' is —'(!<; -^S and therefore the determinant — 192 — is in (I9I12) the factor of D,J,,^^ ^\fn-\-i ^\.-Jn^m- But the factor of this same quantity in '3^"„4-„; is, by inspection, \ / ■"" _L n + lJ « + lV 1 n + »i J m — l „J,n n J n V / — i- mj m + lJ I m + nJ n' Hence it follows from (lOl,,) that ^ + 33^^,qi; f€""-i) ' J. ^ • • • • • • ),i = (-)"+^ -m::^-!^ ± ^^„./^-^„,,/\ ^,„,„/;. 368. By the same j^rocess it will be found that, in general, :r + '33 0]; ^:; g](.'"-i)^f /c' , , ^c^'-d 309. If the flxctor of 93^. in the value of (I9I29) is denoted by ( — y^-kj this expression gives ^ ± ^%% ^ti;r-^^ = ^,(^,213,), in which neither the quantities ('C(;'), nor any function of them, such as K, contain the derivatives of/,,. Hence the derivative of /„, with respect to .r„+i., only occurs in this expression because it is in SB^., in which its coefficient is ^^,_i, so that the term of the preceding expression which contains this derivative is L 'Sli .D^ f \? ,,, is the coefficient of the same derivative in ^ + i>. /i). / B. /, mJ m + 1 ./ 1 • 111 +n J It} the equation (1928) gives H [ ) ^ ^^n-lfljc' — 193 — The comparison of (192y) with this equation gives It is to be observed that, from their definitions, the functions fi^ and 2B^. are both of them partial determinants of the same functions /, /i, /„_i the former being taken with respect to the variables a;,^, ^m + i ^n + m excluding n'„ + i., and the latter being taken with respect to the variables .r, .Tj . . . . .r„_i and a^^+k- In the case, therefore, in which m and n are equal, these two determinants are formed with respect to an entirely different set of variables, and each of the variables x,,^^ is taken in succession from the set x„, x„+i 2:,^ in forming Uf, and combined with the set ^, .Ti ^n-i, in forming 2B^.. 370. The first member of (lOSg) does not contain any deriva- tive of /„ with respect to a variable of which the inferior number is less than m. The factor, therefore, of such a derivative as Z>^/„ in the second member vanishes identically ; whicli is represented by the equation V 371. If in the equation (I9I3) this equation becomes, by writing n — 1 for m, But 25 — 194 — so that if X is supposed to l^e a function of the other variables and / to be equal to x, these equations are reduced to ^, (Q/b,ft-./) = 2,(c^h°'.^) = ^ + ^-,/I^' J5 D'.fl 1 in which Qk = ^,^, ,, and, by (ITGg), — ■'^\ is deduced from Q^ by changing the deriva- tives relating to .r^. into the derivatives relatively to x. This equation is derived from Lagrange. 372. In the greater portion of these formuloB upon functional determinants, the derivative taken Avith regard to either of the variables may be supposed to be frequently repeated, so that Z^^ may be substituted for D^ , and h may even be zero. Thus if, in § 365, D\ is substituted for D^, and if « = 1, the equations of that section are reduced to Hence if and \in is written for wi + 1, the equation (lOOgi) becomes — 195 — If each of the functions (/,) is raultipHecl by /, the values of the functions (e',) remain unchanged, and therefore the value of the determinant ^±fD.^nD.,f, D.,J^^ is multiplied by t''^^. 373. A system of functions (y^) can always be found such that their determinant, with respect to the variables (.r^), may be equal to a given function IT of those variables. For, if all these functions except f,^ are assumed at pleasure, and if f\ represents the form of f„ when all the variables except x,^ are eliminated and the remaining functions (/j) are introduced in their place, the required determinant becomes Hence, /}j is by (187io) determined by the integration in which it must be observed that the quantity under the sign of integration is expressed in terms of /,/i fn-\ ^^^ ^n- In the case of n=i this formula becomes f\ =1 s::, =X 3^.-.. The substance of all these investigations upon determinants is taken without important modifications from Jacobi. — 196 MULTIPLE DERIVATIVES AND INTEGRALS. 374. The functional determinant is shown by Jacobi to be of sinouhir use in the transformation of multiple derivatives and inte- grals. The expression of these functions is facilitated by the notation r\ n-m + l 7^" -»' + ! / ■^fm ■■■■ -^fmJm + l Jn, and If then n = i>;+i TF, a new variable x„, which is a given function of all the variables, /^ may be substituted for either of them as /„ in W, and the new derivative is given by the formula Another new variable :?;„_i may next be introduced instead of fn-\ in the same way, and this process may be repeated of substi- tuting successively for each variable fi a new variable x^, which shall be a function of all the other variables remaining in the derivative at the instant of the substitution of oCi, until, finally, an entirely new set of variables shall be introduced into the derivative. The final form is ^'-DJD.j^ D^j^ = Dl^^W. From the comparison of this form with § 351, it appears that — 197 — the factor of £1 is identical with the determinant of that section. From the reasoning of §§ 353 and 354, it follows that the determi- nant is not changed by substituting in either of the quantities (/) regarded as functions of the variables (.r,) the values of any or all the preceding functions in terms of these variables. But each of the functions (/,) contains, in its present form, none of the succeed- ing functions ; so that, after this substitution, it is expressed in terms of (.r,) . Hence 375. The preceding equation gives, for the multiple integral 'rt + l /•" + ! in which the limiting values of [xi) may be supposed to be constant, while those of (/) may not be constant. If then 77 is determined by the integration so as to contain neither of the variables (.r,) except as they are involved in (/), it is by § 353 unnecessary to have regard to the derivatives of 77 otherwise than as they are dependent upon / in finding the value of the determinant, wdiich is the first member of the following equation, and Avhich therefore becomes ^±D^nD. J.D.J, D.j^ =. ^..Djn= %^S2. Butby (ISOs) — 198 — and, therefore, in which /w«.y denotes that the function to which it is prefixed is referred to the Hmiting values of x,,, so that the difference of the values of the function at these two limits is represented by this notation. But since it is evident from (lOTuj) that JJ lim.,(7r^t) = lim./^" n- and a similar equation may be given for each of the terms of the last member of (1 983), whereby this equation is reduced to J^ /! + 1 pn The multiple integral of the (;z -|- 1) th order is thus reduced to 2ii-\-2 multiple integrals of the nth. order, and this reduction may be continued until the whole process is made to depend upon single integrals, of which one is performed with reference to /, and the number, performed with reference to any other of the variables (/j), is 2'(« + l);« (^ii-\.2 — i). — 199 — II. SIMULTANEOUS DIFFERENTIAL EQUATIONS AND LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 376. An ef|uation /=o, of which the derivative vanishes identically, by means of the simultaneous differential equations represented by Dxi — Xi, in which (X,) are given functions of the variables [xi), is called an iniegral of these equations. It is a general integral if it involves arbi- trary constants, and a imriknlar integral if it does not involve arbi- trary constants. When it involves an arbitrary constant, it is more conveniently expressed in the form f—u, in which a is an arbitrary constant. 377. A function /, which satisfies the linear partial differential equation of the first order is called a solution of this equation. By means of the notation — 200 — this equation may be written 378. The first member of ever?/ integral, expressed in the form (lOOg) or (lOOig), of the simultaneous differential equations (lOOn) is a solution of the fartial differential equation (2OO2) ; and, conversely, every solution of the partial differential equation (2OO2) is the first member of an integral of the simultaneous differential equations (IDQg), and its second member is any constant. For the derivative of (lOOg) or (lOOig) vanishes by the substitution of (lOOn), which gives that is, / satisfies the equation (2OO2). Reciprocall}^, the satisfying of this condition is all that is required in order that (lOOji,) may be an integral of (lOOg). 379. If the equation (199g) is solved relatively to x, so as to express x as a function of the other variables (rr,), the equation ( 19925) becomes X— Fi,„.T = 0, which is distinguished from (2OO2), because the function x, of which the derivatives are taken, is involved in the functions {Xi), whereas /is not involved in these functions. 380. A solution of (2OO2) which shall, for a given equation between the variables, become equal to a given function, may be determined by means of series. For this purpose, let the given equation be (^ = T, in which t is constant, and t a function of the variables, and let the — 201 — solution become a function (p of the variables when this equation is satisfied. If then t were assumed to be also one of the variables of the given equation, and such that in forming the simultaneous equations I)t=l, by which the simultaneous equations become and the given partial differential equation is assume the functional notation n = — f/n, and the integral of the partial differential equation with reference to i is /— 9 — □/=o. which gives f=T^ = (1 + D + n= + etc.)'; This value of / taken from Cauchy, expresses a true solution of the given equation if \Z\'(p is finite for all values of i and vanishes when i is infinite, which is always the case for sufficiently small values of i — t . 381. There are n independent soluiions of the partial differential equation (2OO2) and no more than n indejjendent solutions. 26 — 202 — First. The equation (2OO2) lias n independent solutions. It lias been proved in the preceding section that it has one such solu- tion. Let it then be assumed that m such independent solutions have been obtained, denoted by /„, /„_i /„_,„+i. These independent solutions may be substituted for the m variables, X,,, x„_x ^n-m+i^ with regard to which they are independent ; and if /a/ denotes the value of / when expressed in terms of the new variables, the equations of substitution are represented by in n — »;j + 1 But since the substitution of these equations in (2OO2) reduces it to in Avhicli the functions /t may be regarded as constant. This reduced equation has, then, a solution by the preceding section ; its solution does not involve the variables ^„, ^,i_i .r,i_,„-|-i, and is independent of the given m solutions. The given equation is then proved to have another solution independent of the given solutions ; and this number may again be increased by the same process, until the n independent solutions are obtained, /i,/2 /„. Secondly. The equation (2OO2) cannot have more than n inde- pendent solutions. For if there are {ii-\-Y) solutions (/,), each gives an equation represented by which may be regarded as a linear equation between the quantities — 203 — (X;). By the usual process of elimination, if 7)1,^ denotes the functional determinant of (/) with respect to the variables (x^), these equations give, by § 340, But all the quantities X^ do not vanish, and, therefore, «, = o, or the (n -\- 1) functions (/,) are, by § 355, not independent of each other. 382. It is evident, from the preceding demonstration, t/iat any function of the solutions of the linear iiartial diffe^'erd/ial equation (2OO2) *'^ itself a solution of that equation. 383. A system of finite equations, of which the derivatives are satisfied by the simultaneous equations (lOOis), i;^ called a system of integral equations of the simultaneous differential equations. This system is said to be genercd, when, by the successive elimination of the con- stants, it can be reduced to a form, in which each equation involves an arbitrary constant not included in the other equations, and it is complete when the number of finite equations is equal to that of the given difierential equations. When reduced in the method just proposed, the general system is represented by in which the functions (f/,) are independent of the arbitrary con- stants {[^i). The ixirticular system is represented by a set of similar equations, combined with other equations, which involve no arbitrary constants, and which are represented by i^^^O. — 204 — 384. Each equation of a general sf/stcm of mtcfjral equations, reduced to the form (20824), is an integral of the given sinmltaneoiis differ- ential equations. For the derivative of (20824), when reduced to a finite equation by the substitution of the given differential equations, is independent of the arbitrary constants (fij), and vanishes, there- fore, independently of the equations themselves in which these con- stants are involved. When the system is general, therefore, the functions (g),) are functions of the solutions (/^) of the partial differ- ential equation (2OO2). 385. If the system is particular, and if the number of the equations (2O831), which are free from arbitrary constants is m — n, the same number of variables can be eliminated, by their aid, from the functions (A") and (9^,). The equations, to which (203ai) are thus reduced, are integrals of the simultaneous differential equations, represented by Dxi — Xi, in which the variables (x,), of which the number is m, are those which are not eliminated from (X,) and (9),). 386. The system of equations (2O831) is, by itself, a particular system of integral equations of the given differential equations, which does not contain any arbitrary constant. For the derivative of either of them, involving no arbitrary constant, must be satisfied by means of the equations (lOO^s) and (20831), without any aid from the equations (2O824). The derivative of each of the equations (20834) is, for the same reason, satisfied by the same equations (199i2) and (2O831), without tlie assistance of the equations (2O824). 387. The functions (/) maj^ be supposed to be introduced as the variables instead of the given variables (.r^). By this substitu- tion, tlie proposed system of difterential equations assumes the form — 205 — By this same substitution in the equations (20824) ^"^iid (2033i), the equations (20831) may be readily reduced by processes of elimination to an equal number of equations of the form in which the functions (i^) do not involve those of the functions (/,) of which the values constitute the first members of these equations. Hence the derivatives of these equations, reduced to a finite form by the substitution of (2002) become of the form or x-^ I But this equation does not involve either of the functions (/,) which are not contained in (i^), and, therefore, cannot depend upon the equations (2058). It is, therefore, identical, the functions {Fi) are independent of .r, and the equations (2O831) from which they are derived, contain only the functions (/j). The substitution in (2O831) of the arbitrary constants («,) for the functions (/) to which they are equivalent, reduces these equations to conditional equa- tions between the arbitrary constants. These equations (2O831), there- fore, represoit the conditional equations, to luhich the arl)itrary constants of the integrals o/" (199i2) nnist he sulject, in order that they may coincide with the particular system of integral equations, to ivhich the equations (2O831) belong. After the introduction of the functions (/), instead of the variables {x^), into the functions ((/,), these functions ((f,) can, by the — 206 — substitution of (2058)? be freed from all the functions (/) whicli are not contained in (i^). The derivatives of ((/),) when thus reduced become, by means of the equations (2002), of the form Dcp, = XD^(p, = 0, which must vanish independently of the equations (2058), ^'^^^y therefore, the functions {(pi) do not involve .r. Hence, hy the siihstitidion of {x^ for (/i) the equations (20824) 9^'^^ i^^^ values of {fi^ in terms of {0,^). 088. From any one given integral equation, denoted by z« = 0, the whole system of integral equations, to which it belongs, can be readily obtained. For the finite equation, to which the derivative of this equation is reduced by the substitution of the given differen- tial equations, is, from the very nature of the problem, another of the required system of integral equations. The derivative of this new equation gives a third integral equation, and the continuation of this process leads to the final ddcrmmaiion of the ivholc of the required system of integral equations. 389. This process of deriving a system of integral equations from one of its component equations, affords the means of testing a proposed equation, and ascertaining whether it be an integral equa- tion. For as great a number of independent integral equations is not admissible as that of the variables themselves ; if, therefore, the application of the process to a proposed equation conducts to a numher of indcjyeiideni equations equal to tJiat of the variaUes, it is a sufficient proof that the p>roposed equation is not an integral equation. 390. When a system of integral equations contains superfluous arbitrary constants, that is, constants, wdiicli remain in the functions (^j), after the system is reduced to the form given in § 383 ; such — 207 — constants supply the means of obtaining other integral equations which are not contained in the given system. Thus if (206n) denotes an integral equation, from which the proposed system may be sup- posed to be derived, so that, reciprocally, this equation may be derived from the proposed system, and, therefore, in which F is any arbitrary function ; and if the notation is adopted in which arbitrary constants are denoted by (;',) ; the equation is also an integral equation. For the equation l)u = gives, by direct differentiation, nr^^u — o. But it is obvious, from the form of (2076), Hi^t the derivatives of u with reference to those of the constants (f:?,), which are elimi- nated from the functions (y,) and to which these functions are equal, are, themselves, functions of ((/),• — /i?,) and i/'i ; whereas the deriva- tives of u with reference to the superfluous independent constants (/?,), which are contained in the functions (9,), are not merely func- tions of ((pi — /:?,) and i/^^. Hence the integral equation (207i3) is a new equation, if it contains the derivative of w with reference to either of the superfluous constants (fi,), and there are as many of these new equations as there are superfluous constants. But the number of independent integral equations thus obtained, is, of course. — 208 — subject to the condition, that it cannot exceed the number (n) of the independent solutions of the equation (2OO2). 391. Of all systems of integral equations, that, in which the arbitrary constants are the values which the variables themselves assume for a given value of one of them, deserves especial consider- ation. To simplify the discussion of this case, and place it in the position, in which it will best illustrate the problems of mechanics, the variable (.r), of which the value is given, may denote the ^inie, and the given time is the epoch or origin, at wdiich the elements of the system of variables are given, and from which the variations are estimated. The values of the variables at this beginning of time may be termed tlieir iuifml values, while those at any subsequent time are their Jiual values. The differential equations express the laws of change, under which the variables pass from their initial to their final values, and are equally compatible W'ith any proposed combination of initial values. The initial rallies are, therefore, ivholly arhitrary and independent. Their number is equnl to that of the variables [xi), and, consequently, equal to the tvhole number of independent arbitrary constants, luhieh is required for the complete integral equations. The epoch is also arbitrary, and seems to introduce an addi- tional arbitrary constant. But this constant is obviously superflu- ous ; it corresponds to the arbitrary position of the problem in time, without involving any modification of the essential conditions ; and is the complement of the arbitrary element, wdiich is not expressed, and in reference to which the derivatives in the equations (IQOia) are supposed to be taken. 392. The passage, down the stream of time, from the initial to the final values, conformably to the conditions of change expressed in the differential equations, may be imagined to be reversed and, in a retrograde transit, the same laws of change would, by their reverted action, restore the variables to their initial values. In the — 209 — direct action, the initial values constitute the cause, and the final values are the effect ; whereas, in the reverted action, the final values become the cause of which the initial values are the efibct. Hence it follows that, in any integral equation hetiveen the final and the initial values of the vanahles, the final and initial values of ejich vanahle may he mutually interchanged, and the resulting equation, if not identical iiith the given equation, is a new integral equation. In making this change, the sign of the variable, tvhich expresses the interval of time, must he reversed, because the interval, which is positive with reference to the initial epoch, is negative with reference to the final epoch. If, indeed, the interval were expressed, bj means of the initial value [x^) and the final value {x) of the time, in the form [x — Xq), its sign is directly reversed by the mutual interchange of the initial and final values, which transforms its expression to (.Tq — .r). 393. Let F^ denote the form, which any function F of the final and initial values of the variables assumes after the mutual interchange of these values ; and let represent the system of integral equations reduced so that the functions {(fi) do not involve the initial values (>t'f). The inter- change of the initial and final values in this system, produces a system of integral equations in which each variable is expressed in terms of that one variable, which represents the time, and of the arbitrary constants which are the initial values of the variables. This new system is represented by 394. The discussion has, hitherto, been limited to differential equations of the first order, but it can, readily, be extended so as to 27 — 210 — embrace those of higher orders. If, for instance, the equations are friven in the form o in which the functions (Xj) may involve all the derivatives of the variables (.r,), which are of an order inferior to {pi), each of these inferior derivatives may be regarded as an independent variable, exj)ressed by the form With this new system of variables the given equations are replaced by the differential equations of the first order, represented The number of these differential equations of the first order is easily seen to be {^iPi -|- 1) • 395. When the differential equations are not given in the normal form (2IO3), they can always be reduced to this form. For this purpose, each of the equations, which contains none of the highest derivatives of the variables, must be differentiated as many times, denoted by «•, as are necessary to raise it to an order, wdiich contains such derivatives. If the given equations are represented by ^ = 0; the equations, which are thus derived from them, may be expressed — 211 — in which «, is zero, when it is applied to an equation which is not difFerentiatecl, Each of the derived equations contains at least one of the highest derivatives of the variables, which may be expressed bj Ift'^^-iXi. The functions (9:,) should be independent functions of these derivatives ; whenever this is not the case, such derivatives can be eliminated from the derived equations, and one or more resulting equations will be obtained in which they are not involved. The independence of the functions (9^,) can, however, be directly tested by means of their determinant (18029), which vanishes when it is taken with respect to quantities, for which these functions are not independent. When the fwiciions (9,) are independent wiih resjject to the highest derivatives contained in them, the reqidred normal equations (2IO3) ^-^^'^ obtained from the given equations and their successive derivatives of an order not higher than those of the derived equations (2IO31) Ig the usual process of elimination. For, First, there is a sufficient number of equations, because the number of equations, added to the given equations by differentiation, is 2^i0.i Avhich is the same with the number of derivatives, superior to the order (/>,), the highest of which are to be retained in the normal equations. Secondlg, these equations are independent of each other in respect to the derivatives of the order {pi), and of the superior orders, and, therefore, sufficient for the required elimination; because if any of the equations of the inferior orders were not independent, their derivatives, which are included in the group, (2IO31) would not be independent of each other. 396. When the functions (9)^) are not independent with respect to the highest derivatives contained in them, each of the equations of an inferior order, obtained from the derived equations by elimination, can be substituted for one of the derived equations, — 212 — which is necessarily involved in the elimination by which the reduced equation is obtained. If, therefore, one of the given equations is involved in the elimination, the order of the given equations is reduced by the substitution of the given equation. But if all the equations, necessarily involved in the elimination, were derived by differentiation from the given equations ; and if a denotes the smallest number of successive differentiations, by which either of these derived equations was obtained ; the reduced equa- tion is obviously a derivative of the order {a) of an equation, which can be obtained by direct elimination from those of the given equations, which are of an order inferior by (a) to the derived equations, combined with the derivatives of the other given equations of an inferior order. This reduced equation of an inferior order may, then, be substituted for either of the given equations of a higher order, ujoon which its elimination neces- sarily depends. In all cases, therefore, in ivJiich the functions ((fi) are not independent with respect to the highest derivatives contained in them, the order of the given equations can he reduced hg the substitution of an equation of an inferior order obtained bg elimination between some of the given equatiom and the derivatives of others, 2vhich are of an inferior order. 397. That the normal forms, obtained by the process of § 395, are, as it was remarked by Jacobi, those which are obtained with the least complexity of operation, is easily perceived without any attempt at demonstration. It is, also, obvious, by what modes of substitution other normal forms can be derived from these, which are equivalent to them in the aggregate order of differentiation, but differ in the distribution of the derivatives. Thus if either of the functions (X,) is of an order inferior by {q^) to that of the given equations, it is by (5',) successive differentiations elevated to an order which contains one or more of the highest derivatives involved — 213 — in the normal forms. The (5',:)th derivative of the equation (2IO3), after the values of the highest derivatives, given by the normal equations, are substituted in its second member, so that it is expressed in the form may take the place of this equation in the system of normal equa- tions. If then DY~'^i^i' is one of the derivatives contained in {Xi), and if the normal equation (2IO3) is reduced to the form ])lv-'^iX,' = X[', it may take the place of the equation D^^'x,r=-X,. in the group of normal equations. By means of ( 21814) and its derivatives of an order inferior to the (*/,) th, all the other equations may be reduced so as only to contain derivatives of {xi/) of an order inferior to the [pi, — !7,)th. The nominal system is hj iliis means trans- formed to another normal system, in ivhich the highest derivative of one of the variables is increased, just as much as that of another of the variables is decreased. 398. The repetition of the process of the preceding section may be so conducted that one or more of the variables shall finally disappear from the system of normal equations, and the number of equations will be simultaneously diminished to the same amount as that of the variables. The process may be continued, indeed, until only two variables remain, one of which is the variable (/), with respect to which the derivatives are taken ; but the reduction to this form involves the greatest prolixity and complexity of computa- tion. There are special cases, however, and particularly that of — 214 — linear differential equations, in wliicli this mode of reduction is peculiarly advantageous. The principal portion of this discussion of differential equations is the combined result of the investigations of Euler, Lagrange, Cauchy, and Jacobi ; but an important addition to these researches is now to be developed, for which geometry is eminently indebted to Jacobi. THE JACOBIAN MULTIPLIER OF DIFFERENTIAL EQUATIONS. 399. The function, which was called by Jacobi the neiv multiplier, in order to distinguish it from the Eulerian multiplier, but which, on account of its superior importance, is here distinguished simply as the multiplier of a linear partial differential equation of the first order represented by (2OO2), is that function ivhich, miiltipjlied hj this equation, renders its first member an exact functional determinant ( "Sl^) of the indefi- nite function (/) and of n undefined functions (f) ivith respect to the (?? -j- 1) variables [xi), ivhich are the independent variables of the given equation. On account of the mutual relations of the partial differential equation (2OO2) and the simultaneous differential equations (199i2), this same function may also be regarded as a multiplier of the differential equations (199i2) ; and, for the same reason, it may be considered as a multiplier of the linear partial differ eiitial equation of the first order (2OO20) of n independent variables. 400. If either of the functions (/,), or any function of these functions, is substituted for /, the determinant vanishes, by § 352, and the equation (2OO2) is satisfied. The functions (f) are, therefore, n independent solutions of the equation (2OO2). 401. If the multiplier of the equation (2OO2) is denoted by ^'(0, the condition, by which the multiplier is defined, is expressed by — 215 — the identical equation The equality of the coefficients of Dx.f in the two members of this identity is, by the notation adopted in the theory of determi- nants, expressed by the formula The substitution of this value of ok^ in the equation (ISOa) gives the equation which is a linear 2^(^}'ticil dljf'ercnikd equation of Ihe jird order, l>j ivhivh the multiplier is analytically defined. 402. The defining equation of the multijDlier may by (lOOia) be developed into the form :e,{x,d.^^{^ + ^K^D.x;) = :e,{d,m^d,^ + ^^d^^x^ = o, or This equation divided by ^jait becomes r.log ^((d + ^,D,X, = D log .d(o + 2:,D.X, ■= 0. If all the A'^ariables are regarded as functions of .r, and if x is introduced in place of the element of variation, by means of the formula Dx = X, — 216 — the preceding equation finally assumes the form XD, log ^tD + :E, Z>.^.X, = ; which is an equation involving common differentials, hj ivJiich the miiUiplier is analf/ticallf/ defined. 403. The equation (2158) gives, by (lOlg), when z = 0, the value of the multiplier in the form 404. If the values of (/) are expressed in terms of (x,), by means of the equations (ISOja), and if, by reason of the integrals (199io), the constants {a^ are substituted for (/), the value of the multiplier becomes ^'"" — y ) X' 2:±I)a,F,Da,F, Ba,F^' in which the sign may be rejected at pleasure. 405. In the particular case, in which the equations (189i2) assume the form in which the functions (cj),) involve the arbitrary constants («,), together with no other variable than x, the value of the multiplier is by (ISOia) reduced to ^t = — _— 1 1 XZ±Da^%Da,% DaJP,, ~ X2^+ Da,X,Da,X, Da,X,, __ 1 _ 1 X2:±Df,x^I)j,x, J)f,^x,, — X^}j\,J — 217 — which equdtlon might have been directly deduced from (216ii) and (187io). 406. If the functions (i^) are given independent functions of (/j), they are independent solutions of the equation (2OO2) ^^^ give a multiplier (^ii^) different from '^^ and which is determined by the equation derived from (216n), X^i^i = 2:± D.^F^D^^F^ D.,F,,. This equation, by means of (I8614) and (216n), assumes the form X^{K=^,^^2±Df^F,Df^F, Df^F^ = X^i^ ^±Df^F,Df.^F^ Df,F^, which gives !^= V + Dy,F,Df,F, Dr^F,. The second member of this equation is a function of the func- tions (/,), and may be an arbitrary function of these functions, so that it can have n independent values. The equation, therefore, serves to determine n -\-\ independent values of the multiplier (vj^Xfo^), which is, by (215i2), the whole number of independent values of Avhich it is susceptible. Hence, the ratio of any two multipliers is a solution of the equation (2OO2). It also follows from this argument that every solution of the equation (215i2) is a value of the multiplier. 407. In the particular case, in which Z,D.,Xi^^, one of the n -\- 1 solutions of (215i2) is reduced to a constant, so that in this case, the constant must, contrary to the ordinary usage, be included among the solutions of the equation. The constant may 28 — 218 — be supposed to be unity, and, therefore, one of the muMpUers of the equa- tion (2OO2) is unity, tvhcn the condition (2I727) 2^ fulfilled, and all the other multipliers are solutions of the equation (2OO2). 408. When the solutions (/) of the equation (2OO2) are known, the corresponding value of the multiplier may be determined from (2I611). But it can be derived by a shorter process, when either of the solutions (aa(ti) of (215i2) is known, and also the initial value of q4\ Thus if IT denotes the ratio of 'Jatt. to <£>Ms>, the equation (216n) gives by (194,), When the initial values are substituted in this equation with the notation of § 393, it becomes ok' The value of 77° may, by the elimination of the variables {xf) be reduced to a function of the functions [fi) ; and, if in this expression the functions {f) are substituted for their initial values (/J), the value of 77 is reproduced. For the function, which is obtained by this substitution, is a function of {f) and therefore a solution of the equation (2OO2); and it is, moreover, that particular solution, of which the initial value is the given function 77°. 409. In the especial case, in which the initial values of [f] are the variables {x^), the value of Qk° is obviously reduced to unity and the equation (218ii) becomes 77°=vfc(i:X°. 410. When, in the differential equations (199i2), the arbitrary — 219 — element of variation is assumed to be the variable x, the value of X is unity ; and, in this case, the equation (218ii) becomes ok; ' which in the case of the preceding section is reduced to and when, moreover, the equation (217^7) '^^ satisfied, so that one of the multipliers is unity, this value is still further reduced to 17° =1. 411. The arbitrary constants («,) may be substituted for the functions (/,) in the equation (218n), when it is regarded as result- ing from the integrals of (lOOia). By this substitution U becomes a function of the arbitrary constants, which may be represented by C, and the equation gives, by means of (187io), ^l,n = -^ i Da^XiDa^X^ Da^X^ = ^k ■ X' The logarithm of this equation becomes by the substitution of (2I62), and including C in the constants of integration, log -S- + I)a,X^ I)a,X, Da„X, = log^ + log C— log <^i^, in which all the functions (X,) can evidently be multiplied by any common factor, without disturbing the equality. 412. In the especial case of X=l — 220 — the preceding formula becomes 413. When simultaneous differential equations are transformed from one system of variables to another, the multiplier usually under- goes a change at the same time, but there are conditions, to which the arbitrary element of differentiation may be subjected, and under which the multiplier remains unchanged. Thus if the new system of variables is represented by (tVi), if the equations (199i2), in their new form, are represented by in which the accented sign of differentiation refers to the new arbitrary element of differentiation, and if -jj — ^y the values of ( Tl^) become, by (lOOa^) and the preceding formulae of this section. This value of ( TF^), in combination with the formulae (lOOgs) and (2162), gives ^G^, {X.D.J) — 221 — If oN* is a multiplier of (22O12), the defining equation of (2102) is, in respect to this multiplier, cN'^,( W.D.J) = 2 + DJD..J, i?.„/„. The ratio of the equations (22O27) ^"^^ (22I3), reduced by means of (I8613) and (187io), gives ^ oir _ 2±DJD.J , ZJ.„X = -2" + D^^xDxr^Xi Du-^x^ = {^± DjvD.^iv, D^w,,)-\ If, therefore, the multipliers oV and <£A^ are equal, the value of G becomes G', if G' =^ ^ + D^,xDw^Xi Dii-n^n = {:E± D,ioD.^iL\ D.w:)-^. 414. The equation (21525), applied to the new system of varia- bles (?6',), gives, by means of this equation and (22O17), if the multi- pliers are, for the instant, assumed to be equal, ^,Z>„. TT^ = — i>' log ^(t = — 6^'i> log ^tt = g':e,d.^x,. 415. If the arbitrary element of differentiation is supposed to be the same in both systems of variables, the values of G, Wi, and oN* become G=l, c^=G'^Ais. — 222 — 416. If the first m -f- 1, only, of the variables (:r,) are exchanged for the new variables {u'i), which limitation is expressed by the formula the value of G^ is abbreviated to G' :=:S± D^xDu^^Xi -Z>«.,„rr^ = {2±D^wD.^iv, i>.„.«^.)-^ 417. Hence if the arbitrary element of differentiation, com- mon to the two systems, is one of the variables and is expressed by t, so that the remaining variables are still denoted by (x^) and (tt\)y the formula (22I15) continues to express the value of G\ 418. If the last {n — m) of the variables {n\) are solutions of the equation (2OO2), the corresponding values of the functions ( TF|) vanish by (22O22). If the multiplier is also supposed to remain unchanged, the partial differential equation (2OO2), by which it is determined, is reduced to 'J The arbitrary constants (/:?,) may, therefore, be substituted for the solutions (?t',), and the value of G' becomes 419. But if, instead of the equality of multipliers, the ele- ments of differentiation are identical in the systems, the defining equation is expressed in the slightly different form of — 223 — in which the functions ( TT',) and the muUipher (oS') are given by (221»). 420. If the variables (tVi) which are retained, coincide with the original variables (x^), the equation for the multiplier becomes m in Avhich By the formulae of this and the two preceding sections the multiplier of the system of differential equations, to which a given system is reduced by means of any of its integrals, can be obtained from the multiplier of the given system. This will, soon, appear to be one of the most important properties of multipliers. 421. If the given differential equations are of an order, which is higher than the first order, and have the normal form (210g), the equation (21525), by which the multiplier is defined, is simplified by the consideration that > The multiplier of the given equations, or of the equations (2IO15), ^^ which tliey should he replaced, is, therefore, determined hy the equation B log ^((d -f ^ .Z>,tP -1) Xi = . 422. If the functions JQ do not involve a:/^"^' or if, in general, unity is one of the values of the multiplier of the given equations. 224 423. If the given equations have not the formal form, but have the form such that they involve no derivatives of a higher order than the nor- mal forms, to which they are reducible by immediate elimination without differentiation, the equation for determining the multiplier assumes a simple symbolic form, by means of the notation For it is to be observed that each of the subsidiary terms, of which the second term of the equation (22825) is the aggregate, is to be obtained from the equations (2243), by taking their derivatives relatively to a^^r'^^ on the hypothesis that a^^k* are functions of this variable, and thence determining, by elimination, the values of these subsidiary terms. Hence if the derivatives of (2243), relatively to x^l'r'^^ are represented by ( 17724), provided the letters t of that equation are accented i times, and the number k is written below the u. From the comparison of (ISOig) with (224ii), it appears that {i,}c) vanishes in the present case, and that the sign of d is to be reversed, whence the equation (I8O26) becomes The equation (22825) ^^ which the multiplier is determined, assumes the symholical form D log ^ti = _ :^,m — (T log 'Sto^. — 225 — 424. It may, sometimes, happen that the vakies of a^[^ and ^.«i,'* are such that the sum of da^l\ and of lDa^/^\ in which I is constant, is simpler than da^^K In this case, if d'^d + in, the addition of to the equation (2243i) gives the sf/mhoUcal form i>log(.ci(tQj5^j^) = (5qog^.,. 425. If the given differential equations have the form (2IO27), so that they cannot be reduced to the normal form without differ- entiation, the equations (2IO31), wdiich are derived from them by differentiation, give, by direct elimination, a system of normal forms, which include, as a reduced system, the normal forms finally obtained by the process of § 395. The multiplier of the equations (2IO27) is determined by the symbolic equation (22*43i), or (225io), provided that in the values (224io) of a^^ and da^l'^ from which ()^'3l„ is consti- tuted, the value of /?^. is increased by a^. 426. The values of «^' and (5'«SJ* may be determined directly from the equations (2IO27). For this purpose, if X is written instead of a in order to avoid the confusion which might arise from the use of a as an arbitrary constant, and if the ingenious notation, which is familiar to the German mathematicians, for the continued product of all the integers from 1 to X inclusive. "O" l\ = l{l — l){l^2) 3.2.1^ is adopted, the equations (2IO31) are represented by 29 — 226 — and we find, by well-known formula?, D^^K) 9) = 1"^ D} {p^M FD,(K) :?;''■'] V ' ,/'L(x-r)!(;.-. + ,.)! ' ^ J The inferior limit v is determined by the condition that neither V nor X — yi A^ v' can be negative. Hence if X + 1>K, /==0, if ^_l0, the value of the factor of G is — 228 — but if this factor is The logarithm of the complete value of G is, therefore, log 6^=:^. [(?, + !-?,) log ^S]. PRINCirLE OF THE LAST ML'LTIPLIER. 428. The consideration of the case in which there are two variables, leads to a valuable principle of integration, discovered by Jacobi, and which he called the principle of the last multiplier. In the case of two variables, the equation (2162) becomes which gives Hence it is obvious that Df, = ^(t {XDx — X,Dx\) or, by integration. /i= C^{\^{XDx — X^Dx^), so that ivhen the multiplier is hioivn, this equation determines the integral of the ttvo differential equations (199i2) of tivo variables, or that of the sin- — 229 — gle equation to which they are eqidvaleiity and the miiUijjUcr is, in this case, identical iviih the tvell-hiown Eulenan multiplier. 429. ^Yhcn all the integrals hut one of a given sgstem of dij^erential equations (199i2) are hiown, of ivhich the multiplier is also given, the last integral is determined hy quadratures ly the ^^rocess of the preceding section ; because the multiplier of the two differential equations with two variables, to which the given system may, in this case, be reduced, is determined from the given multiplier by § 418. This is Jacobi's principle of the last multiplier. 430. In the case of § 380, in ivhich the element of variation [t) is one of the variables, if the functions {Xi) do not involve it), the equation (2018) gives i from which t can be determined by quadratures, when all the other integrals of the given equations are known, even if the multiplier is not known, provided that Xj is reduced to a function oi x^, by means of the known integrals. If the multiplier is also Jcnown, and if it does not involve t, the last of the integrals ivhich do not involve t can he determined hy the process of the preceding section, and, therefore, the two last integrals of the given equations can, in this case, he determined hy quadratures. Bid if the given multiplier (^(t) involves t, a mxdiiplier {ysMs^^, ivhich does not involve t, can he derived from all the integrals ivhich do not involve t, and the quotient of these two multipliers gives hy § 406, an integral involv- ing t, and ivhich taJces the place of (229i7) ; ^^ ^^^^<^? ^'^^ ^-^"'^ ^^5^, the last integral is determined in a finite form without integration. — 230 — 431. This proposition was shown by Jacobi to admit of the following generalization. If all the functions {X^), in tvhich i is greater than m, are free from those of the variables [xi) in ivhich i is not greater than m, and if the remaining functions satisfy the equation m * two integrations can always he performed hy quadratures, whenever a multi- plier is hioivn tvhich does not involve the variables (:?^,<^ + i), hut when the given multiplier does involve either of these variables one integration can he performed hy quadratures, and another integral is given, immediately, with- out any process of integration. For if the given multiplier <£J^ involves only the variables (.^^>^), it not only satisfies the condition (215g), but also on account of the equation (2306) and is, therefore, a multiplier of the portion of the equations (199i2) in which i is greater than m. This portion of the given equations can, therefore, be first integrated, independently of the remainder of the system, and the last integral of this portion will be obtained by quadratures, because its multiplier is given. But the last inte- gral of the whole system may, also, be obtained by quadratures, because its multiplier is known; so that two of the integrals can be obtained by quadratures. But if the given multiplier involves any of the variables (^i<»n + i)> the separate integration of that portion of the equations (199i2) in which i is greater than m, gives a multiplier of this portion involving only the variables (.'r,^,^), which satisfies the equation (23O15) J and by (2306) it also satisfies the equation (2158), so that it is a new multiplier of the given equation. The quotients of these two mul- — 231 — tipliers gives, by §406, an integral involving (.r,<-,„^i),and which takes the place of the first of the two integrals, which are obtained by quad- ratures when the given multiplier involves only the variables (:i\>,„). PARTIAL MULTIPLIERS. 432. Additional to the systems of Eulerian and Jacobian mul- tipliers, and inclusive of them, are those, of which I have given the investigation in Gould's Astronomical Journal, and which I have called partial inuliipliers. The partial multipliers of the differential equations (199i2) are represented by (^si^il^i^,,^,,^^ ^.J, in which i,ki,k2,. . . etc. are any different numbers, or by (^'^'f^/,A-)j ii^ which I and K denote groups of numbers ; and they are defined by the equation f<£i^^j=:e±d ad f, DA, in which P is any arbitrary function, /"i , /"a • • • hi are numbers not included in the groups I, and /u/a, etc. are solutions of the equa- tion (2OO2). The notation (o&lU^') may also be used to denote the multiplier, with the definition that if ^denotes the group of numbers represented by (/",„). 433. The system of multipliers of (lOOja), evidently, satisfies the system of differential equations, which are derived from (187io), and represented by in which i includes all the numbers not belonging to the group I. 434. The group of all the numbers not included in the group — 232 — (/) with the exception of any two, which may be selected at pleas- ure, may be denoted by H. The elimination of the corresponding values of X/. from the equations, obtained from (2OO2) by the substi- tution of the various values of (/) gives the equations, which are represented by This system of equations combined with that of (23l2s) defines, analytically, the sj^stem of partial multipliers. 435. In the formation of the multipliers, a careful regard must be had to their signs, conformably to the rule of formation of deter- minants, so that in general 436. In the special case, in which the group (/, /) of §433 is reduced to a single number, and in which P is X, the preceding equations become = ^,D^^ (X^((o,) = :^,D^^ (a:,(t X,) ; so that, the miiUij)l'ier is, in this case, the Jacohian multiplier. 437. In the case, in which the groups {i,I) of § 433 include the numbers of all the variables but one, and in which P is unity, the equations become I k so that, the system of multipliers is, in this case, that of the Eulerian multi- pliers amplified hy Lagrange. — 233 — 43S. The partial multipliers may be denoted as ihc frsf, second, etc., to the lad corresponding to the degree of the determinant which is the second member of the equation (216n)- With this designa- tion, the last multiplier coincides with the Jacobian multiplier and gives a last integral of the differential equations, while the first mul- tipliers coincide with the Eulerian, of which each system gives a first integral of those equations. This proposition may be general- ized, and it may be shown that each Si/stem of muUijjliers determines an integral of the given equations ly means of quadratures, and holds a place in the rank of multipliers similar to that held hy the integral, in the rank of {integrals. The investigation of the relations of the multipliers of differ- ent systems will be found to lead immediately to this proposition, after its truth has been established in the case of the Eulerian mul- tipliers. 439. The deduction of an integral of a system of differential equations (199i2), by means of quadratures, from a given system of Eulerian multipliers, is quite a simple process. For the definition of these multipliers in § 437 gives Df^^,UK\^^Dx^. If the quantities represented by ( Q^) are defined by the equa- tion Ox, ' the required integral is f=^,q^ = a. For the defining equation of Q^ gives ' 30 — 234 — Hence it is found by differentiation that ( Qi) is free from all the variables (%<,), for if this is supposed to be proved for ( Q;,<^i) it it seen, by (23227), that hi fi *0 The differential of (23337) i^? therefore. • which corresponds to the required differential (233i9). 440. When the differential equations (lOOja) are transformed to other variables in the manner which is indicated in § 413, any multiplier of the new system is obtained by the following formula which corresponds to (23I15), P'd<^=Z± D^^f, D,,^f, D,,J,, . If, then, the functions {G) are defined by the equation =z{Z + D tvu D^ IV, D^ tuj, )-^ 1 2 'ill '" the proposition (I8620) gives by (23I15) P^Q>r^=P^^(^^G^(/f)). 441. If any of the solutions (/) of (2OO2) are known, they can be assumed as new variables to take the place of either of the given variables, and the new multipliers must be determined by — 235 — the preceding equation. But it is evident tliat, in this case, the number of elements which compose each of the terms of (cV^) will be diminished by a number equal to that of the solutions, which are introduced as variables. Hence since ?)i is the number of ele- ments which compose each term of (oE^ttj), if [m — 1) is that of the known solutions, the number of elements of (g-V^) may be reduced to one, in which case the multipliers (g^h) become Eulerian and give the mih solution of (2OO2) or the mth integral of (IQOja), by means of quadratures, which corresponds to the proposition of § 438. III. INTEGRALS OF THE DIFFERENTIAL EQUATIONS OF MOTION. 442. When the differential equations of motion are expressed in their utmost generality, there is no known integral which is suf- ficiently comprehensive to embrace them. But the equation (163i4) of living forces is an integral, which is applicable to all the great problems of jihysics, and holds the most important jDOsition in refer- ence to investigations into the phenomena of the material world. There are other integrals of great generality, which might be inves- tigated in this place, if they were not clothed with such a character of speciality, that they properly belong to some of the following chapters. The application of Jacobi's principle of the last multi- plier to dynamic equations gives results of so general a character, that their investigation cannot appropriately be reserved for any chapter devoted to the consideration of special problems. — 236 — tiik application of jacobl's principle of the last multiplier to Lagrange's canonical forms. 443. It follows from the homogeneous nature of T (165io), that each of Lagrange's equations (164i2), involves one or more of the quantities represented by (>/'), and the system of these equa- tions has, therefore, the form represented by (2IO30). If, then, (a^') denotes the coefficient of (i^^) in the value of (tu^), given by (IGoj), this value becomes and that of T is by (165n) so that the functions [af) only involve the quantities represented by (1]) and the time (/), and satisfy the equations 444. Each of Lagrange's equations may be expressed in the form i i Hence, when £1 is only a function of (j^,) and t, the equa- tions (224io) become D rr(fi — af, k _ 237 — from which are easily derived the equations k i The notation k i gives [i,k)= — {Jc,{), dd^ = D,a^^ + {i,Jc). In the substitution of these values in (2243i), it is evident from (ISOig), (I8O31), and (ISlg) that the functions {i,Jc) disappear, and since D takes the place of i>,, (2243i) becomes i>logaB.((er=:i>log^^„, and, therefore, since the arbitrary constant may be neglected, which holds, even if the equations of condition involve the time. In all dynamical problems, therefore, in ivhich the forces are indepen- dent of the velocities of the moving hodies, a Jacohian multiplier is given directly by the equation (237i9), so that the last integral can always be obtained by quadratures. 445. Hence, by § 430, in any dynamical problem, in ivhich the forces and equations of condition are independent of the time as tvell as of the velocities of the bodies, the two last integrals can be obtained by quad- ratures. 446. The substitution of hi = Xi\J m,Ui + i — l/i\l i^h^'Ui + i^' ^.\l m„ — 238 — in (lG42o) and (16228) gives 2 2 2 Hence if the value of w^^'^ is by (23G15), which, combined with § § 346 and 348, gives in which 'S^*/*'' denotes the functional determinant of a group {M) of {n -\- 1) of the functions (^"i) relatively to the variables [r]i). It may be observed that if n^ is the number of bodies of the system, and n^ the number of conditional equations, the value of n is « rrr 3 n^ t^ 1 . 447. If the conditional equations are represented by and if k their derivatives with reference to [iji) are represented by If then (H) denotes any group of n of the quantities (^",), and — 239 — {n,h) denotes a group of 72-(-l of the same quantities in which the group [H) is included, the preceding equations give, by elimination, between all those in which i remains unchanged, Since then the group {H,h) is also denoted by (oeXId), if the group of all the remaining quantities (ij) is denoted by {N), '\? M' and N' are other groups of the same species, and if ( ^'^^) denotes the determinant of the corresponding values of (4/')> ^^^® preceding equations give, by elimination, n n i which, it is easily seen, may be extended to the case of any groups whatever (iltf and Mi), in which each includes {n-\- 1) of the quan- tities {li). If, therefore, some one group is arbitrarily selected and denoted by {M^i), the equation (28813) becomes 448. If the derivatives of (1],) relatively to (^,) are denoted by and if ^^^^ denotes the determinant of the values of {e^h), which cor- respond to those of (%"') in ^^n^\ the derivatives of (1^,) may first be taken with respect to [l^), and if those of (^\) are afterwards taken with respect to {i]i), they give by (I8620) — 240 — Hence, if gN* denotes the determinant of all the quantities {TFi) and ()/,) with reference to (^,), the equation (239i3) gives which; substituted in (2392o) reduces it to 449. If there are no equations of condition, the value of ^slUs is reduced to = (^ + i),£/),;S D,Lf 1 71 1 n If in this case, therefore, the values of (i]i) coincide ivith those of (^i), the multiplier is reduced to unity. 450. If the equations of motion were given in the system of § 310, in which the forces, represented by the equations of condition, are included in those of i2, this system might, by means of the equa- tions of condition, be reduced to that of Lagrange's canonical forms. In performing this reduction, the equations of condition hold the same relation to the differential equations, which the equations (2IO27) hold to the equations (2IO31), in performing the reduction of § § 395 and 425. It is also obvious that i i Hence the divisor by which the multiplier of the first of these — 241 — systems is reduced to that of the last, is by (2283), (222;), and the preceding sections {:e±d^i]D^ lu A nnD. nn^ n, d^ //„ f^djp- 1 n n + 1 n + 2 Zn^ 2 and, therefore, the multipher of the system, previous to reduction, is by (2408) 451. If the system of differential equations is given in Ham- ilton's form, (I663), the equation (21625) ^^^ the determination of the multiplier becomes D log .dt + 2:, {D^ D^- D^D,;) 11^,^ = i>log ^ik = 0, i i i i whence the multiplier of this system is unity. CHAPTER XI. MOTION OF TRANSLATION. 452. If the coordinates of the centre of gravity of a system are r^-^j^^, ^g, and if those of any other point axa Xg-{-Xi,y^-\-7/i, Zg -\- Zi, the value of T becomes, by (16228) ^^^ (1642o) and the con- ditions of the centre of gravity (155i9), r=i?^^,m,4-^^,(^;^,e'f). 31 — 242 — Hence the motion of the centre of gravity is determined by the equation, derived from (164^2), :^,?«, A.< = ^,mMx, = D, n, s and the corresponding equations for the other axes. The value of £2 may be restricted in this equation to the external forces and those which correspond to the external equations of condition, for the internal forces and equations of condition being dependent solely upon the relative positions of the bodies of the system, are functions of the differences of the corresponding coordinates of the bodies, from which Xg,^g, ^g disappear. The motion of the centre of gravity is, therefore, independent of the mutual connections of the ^mrts of the system, and is the same as if all the forces toere applied directly at this centre, provided they are unchanged in amount and direction. 453. Since the second member of (2423) expresses the whole amount of force, acting upon the system and resolved in the direc- tion of the axis of x, this equation expresses that the motion of the cen- tre of gravity in any direction depends upon the whole amount of external force acting in that direction. If, therefore, the ivhole amount of external force acting in any direc- tion vanishes, the velocity of the centre of gravity in that direction is uniform. MOTION OF A POINT. 454. When the system is reduced to a single point, it becomes a mass united at its centre of gravity, and the only possible motion is that of translation. The position of the point is determined by three coordinates, which, combined with their derivatives and with the time, constitute a system of seven variables, and require, in gen- eral, six integrals for the complete determination of the motion of — 243 — the point. The differential equations become, in this case, if the mass of the body is assumed to be the unit of mass, with the corresponding equations for the other axes. A POINT MOVING UPON A FIXED LINE. 455. The two equations by which the line is defined are two equations of condition, which may be denoted by Together with their derivatives, they take the place of four of the integrals of § 454. 0/ the iivo remaining integrals, when il does not involve the time, both can le determined hy quadratures hy § 445. One of these integrals is, indeed, the equation of living forces (IGSig), which becomes in this case The final integral is obtained from this integral by the equation Dr,S 456. It follows from (243i9) that the velocity of a body only depends upon its initial velocity and the value of the potential at each point of its path ; and this conclusion coincides with the propo- sition of § 58. In zvJiatever path, therefore, a body moves from one j^oint to another, the increase or decrease of the square of its velocity may he meas- — 244 — lived hy that of the potential, ivhen the equations of condition and the forces tvhich act upon the system are, like the fixed forces of nature, independent of the time and the velocity of the hody, 457. If there is any point upon the line, beyond which the decrease of the potential exceeds one half of the square of the initial velocity, the body cannot proceed beyond that point. If there is, in each direction from the initial position of the hody upon the line, a lim- iting point of this description, the motion of the hody is restricted to the inter- vening space. Since the body can only have the direction of its motion reversed at the limiting points where its velocity vanishes, it must oscillate back and forth upon the whole of the intervening portion of the line, according to the law expressed by the equation (24323). It is evident from the inspection of the equation (24323), that the time which the body occupies in passing from any point (J.) of the line to another point (^), must be the same with that which it occupied in the preceding oscillation in the reverse transit from the point {B) to the point (^) ; and, therefore, the entire duration of oscil- lation must he invariahle. 458. If the line returns into itself, and if there is no point upon it for which the decrease of the potential is as great as the initial power of the body, the hody will continue to move through the ivhole circuit of the line, and ivill always return to the same point ivith the same velocity, so that the period of the circuit ivill he constant. 459. When the forces and the equations of condition involve the time, the multiplier becomes by (238ig) and the last of the integrals, which are required to solve the prohlem, can he ohtained by quadratures. 245 — THE MOTION OF A BODY UPON A LINK, WHEN THERE IS NO EXTERNAL FORCE. CENTRIFUGAL FORCE. 460. When the line is fixed, and there is no external force, SI van- ishes in (243i9), and the velocity is, therefore, constant. 461. In this case, the line may be regarded as the locus of a resisting force, which acts perpendicularly to the line. The plane of X and y may be supposed to be, for each instant, that of the curva- ture of the line at the position of the body, R may be the resisting force of the line, and (> its radius of curvature ; and elementary con- siderations, combined with the equation (I6425), give DtX = Z>,5 sin ^ = z; sin ^, D^X := V cos^'Z>^^ = v^ cos'^D/^ = - cos^ = ii cos'^, whence Q so that the j^^^ssnre against the line is measured hy the quotient of the square of the velocity divided hy the radius of curvature, which is called the centrifugcd force of the body. 462. If there are external forces, the whole pressure iqjon the line is ohtained hy comhining the action of all the external forces resolved perpen- dicidarly to the line, with the centrifugal force. 463. The centrifugal force cannot be used as a motive power in machinery, for the body moves perpendicularly to the direc- tion of this force ; and, therefore, the power communicated by it vanishes, because it is measured by the product of the intensity of the force multiplied by the space through which it acts. 464. If the line is not fixed in position, but has a motion of — 246 — translation, the same motion of translation may be attributed to the axes of coordinates, so that the coordinates of the moving origin at any time may be a^., Uy, a,, with reference to the fixed axes. If the coordinates of the body with reference to the moving axes are b^? ^ l^, the value of 2 ^T (1642i) becomes 2r=^,(,^;+«;)^ =r DtS'^ -\-2wI)iS COS w-\-^y 2 ^=s' -{-22vs' COS ,; -|- iv"^ if 2v denotes the velocity of the motion of the origin, and 5 the length of the line passed over by the body. Hence Lagrange's equation (164i2) gives Dt (/ -f- tv cos /„) = Ds (/ zv cos ;',) == / tvD^ cos ^ . But, since the angles which 5 makes with the axes are inde- pendent of the time, the derivative is A («f cos ^,) = D,:E^ (tv^ cos ■;) = 2^ (?< cos ^ + / tv^ D, cos i) = Z^ {w^ cos %) -\- / D, {vj cos ,;) , which reduces the preceding equation to J),s = — 2:, {iv^ cos ^) = — TFcos ^y, if 2 2 2 denotes the acceleration of the line at each instant. Hence it is easy to see that if the acceleration is perpendicular to the line, the relative velocity of the hod// to the line is not clmnged ; hut if the acceleration is in the direction of the line, the change of relative velocity is exactly equal to the — 247 — acceleration, so that there is, in this case, no change in the actual velocity of the hody in space. 465. It follows, from the preceding investigation, that if the motion of the line is uniform, the relative velocity of the body and the line remains constant. 466. It is also apparent from this investigation that even under the action of external forces, the relative motion of the body to the line may he computed, hy regarding the acceleration of the line as a force acting upon the body in a direction opposite to its actual direction. 467. If the line rotates about a fixed axis, which is assumed to be the axis of z, let u be the projection of the radius vector upon the plane oixy, (p the angle which u, makes with the rotating axis of x, and a the velocity of rotation, and the value of2T becomes = (I),sf-{-2u''(f'a-{-u''a'' 2 =^s' -\-2uas^ COS ^ -\- li^a^, in which 6 is the angle, which s makes with the elementary arc udcp. Hence the derivatives of T are D.T^ns -}-««cos^, D, T= s' D,{ua cos &) + a^u cos ;' ; and the equation (164i2) becomes D^,s^= — u cos t1 a -j- a'^u cos ". The former of the two terms which compose the second mem- — 248 — ber of this equation, is the negative of the acceleration of the rotar tive velocity resolved in the direction of the arc of the rotating line. The latter term represents the centrifugal force, which corresponds at the body to the rotation {a), and which is also resolved in the direction of the moving arc. But the centrifugal force is purely relative in its character, and arises from the resistance of the body to accompany the curve in its change of motion occasioned by rota- tion. These terms combined show, then, that in this case, as well as in that of translation, and, consequently, in every case the relative motion of the hod// to the line may he ohtained hy attributing to the hody the negative of the acceleration of the line, ivhich occurs at the position of the hody ; in the case of external forces, their action must he united to that ivhich arises from the acceleration of the line. 468. In the case of an uniform rotation about a fixed axis, the equation (24729) becomes Z^ 5 = «^ II cos " = a^ u D, u . The integral of the product of this equation, multiplied by 2^5, is {D,sf^a\u' + A), in which A is an arbitrary constant. Hence it is obvious that 469. When the constant [A) is negative, the value of u cannot be less than y/ — A; so that when the body approaches the axis, its velocity upon the line is constantly retarded, and vanishes, when its distance from the axis is reduced to y/ — A, after which the direction of the motion is reversed. If the portion of the line, upon which the body moves, extends at each extremity, so as to be at as small a dis- — 249 — tance as y/ — A from the axis, the hody iiill oscillate upon it with a con- stant jyeriod of oscilkdion. 470. When the constant [A) is positive, or when it is negative, and no joortion of the line in the direction, towards which the body is moving, is at so small a distance as y/ — A from the axis, the motion of the body upon the line will constantly retain the same direction. If, moreover, the curve returns into itself, the hody ivill always continue to move around it, with a constant period of revolution. 471. When the constant (^) vanishes, the equation (2482o) gives DiSr=.uU Js U J a " If the curve, also, passes through the axis of rotation, the value of i>„5 may be supposed to be constant, while the body is very near the axis, and may be represented by (i ; so that the motion of the body in the vicinity of the axis is given by the equation a / 1= ^j log u . The second member of this equation becomes infinite when u vanishes, and, therefore, the motion of the hody, in this case, is infinitely sloiv in the immediate vicinity of the axis. 472. When the rotating line is straiyM, let p be the distance of its nearest approach to the axis of rotation, and ^ the angle which it makes with the plane of xy. . If then s is counted from the foot of the perpendicular, which joins the nearest points of the line and the axis of revolution, the 32 — 250 — value of i^ is given by the equation ll" Z=L f ■\- i (IQ^- ^ '^ whence (24804) becomes, in this case, = ^-l^log[.cos^ + V(/+^+.^cos^^)]-^^^|f^; in which the arbitrary constant is determined so that t may vanish with 5, and this equation is apphcable when (jo^ -)- A) is positive. In this case, the substitution of the notation , h tan 9== -, ' S COS d reduces the preceding equation to a t cos ^ = log cot h 9 . But when {^f--\-A) is negative, the substitution of the notation Ti'^ = -{f-\-A), h smif 5 cos 5 and the determination of the arbitrary constant, so that t may vanish when s has its least possible value of Jc sec ^, reduce the equation (2506) to a t cos ^ = log tan ^ 1// . — 251 — When (// -|- A) vanishes, the equation (250o) is reduced to e a t cos 6 = log — ; So in which Sq is the initial value of s. When p also vanishes, the sur- face described by the line is a right cone, and when it is developed into a plane, the path, described hy the hodi/, becomes a bgarithmic spiral. 473. When the rotating line is the circumference of a circle ivhich is situated in the plane of rotation, let R denote the radius of the circle, a the distance of the centre of the circle from the origin, 2 ip the angle, which the radius of the circle, drawn to the body, makes with that which is drawn in a direction opposite to the origin, and the equation (24824) becomes 2R at =/ ^lA-\-{R-\-af—4.aEs.m^(f'\ When A-\-(R — a)^ h positive, which corresponds to the case of § 470, let h^=.A + (R + a)\ AaR BlV^i W and by the notation of elliptic integrals of § 169, the equation (251i8) becomes 2 R rr When i is so small that its fourth power may be rejected, this — 252 — equation gives, by an easy reduction, «zf = (1 -[- 4 sin^2)— y-^ — jT sin^2;sin2g5. In this case, therefore, the time of describing the semicircum- ference, for which 2 cp is greater than a quadrant, exceeds the time of describing that for which 2(p is less than a quadrant by T-siirt h K' ~\_A-\-{R-\- afj ' When A -\- {li — «)^ is nc(/at ive, \v\iich. corresponds to the case of §469, let surt=z- — -, AaJi sin (5 sin(p and the equation (251i8) becomes 2R r , 2Rsmi C at-^^- I sec 6 = — -. — / sec 9) When i is so small that its square may be rejected, the duration of an oscillation becomes ay a When the circumference passes through the axis of rotation, a is equal to 7?, and the time of the small oscillation becomes identical with that of the semi-revolution of the circle; but the time of a larger oscillation exceeds that of the semi-revolution. — 253 — When A-\- (R — a)^ vankhcs, the equation (251i8) becomes When A-[-{R — ciY is very small, and its ratio to ^ciR is denoted by d li, the equation (251i8) gives throughout the greater portion of the path, in which (f difiers sensibly from ^ n, that is, in which the body is not near its point of closest approach to the axis of rotation, so that the square o^ ^ A may be neglected, = (1 — ^d A) i/— log tan (j ti -\- i(p) — }dAi/~t(in(psec(p. But in the vicinity of the point of nearest approach, let If = in — (p be so small that its square is of the same order with d A, and the equation (251i8) gives at=z — v^- I .-,,... = — v^ - Sin^~^^ -=, when d A is positive. = — i/— Cos^~-^^ , - ' ., , when dAis neo;ative, 474. When the rotatinc/ line is ivholly contained upon the surface of a cylinder of revolution of zvhich the axis is the axis of revolution, u is con- stant and the equation (24729) becomes D.sl)^s=^ — u^a , from which (p or s may be eliminated by the given equation of the curve. — 254 — 475. When the velocity of rotation is constant, the second member of (25328) vanishes, and the velocity of the body is conse- quently uniform. 476. When the curve is a M/o:, the value ofi).s is constant, and the equation (25829) gives in which A is an arbitrary constant. 477. When the acceleration is uniform, a is constant, and the integral of (25328) gives iij^a't^z i --— ; in which A is an arbitrary constant. MOTION OF A HEAVY BODY UPON A FIXED LINE. THE SIMPLE PENDULUM. 478. When the line is fixed, and the force which acts upon the body is that of gravity at the surflice of the earth, represented by ff, and the axis of 2 is assumed to be the vertical, directed down- wards, the equations (243i9_32) give t-c 1 r A^ 479. If the curve is contained upon the surface of a cylinder of which the axis is vertical, the motion of the body is the same as — 255 — it would be upon the plane curve, obtained by the development of the cylinder into a vertical plane ; because the value of D^s is not changed by this development. 480. If the fixed line is straight, the equation (2542s) becomes cos i gt = \l{2gz J^2H) — sl{2H) = v — v,, if Vq is the initial velocity of the body. 481. If there is no initial velocity, the preceding equations become (ioslgt=zs^{2gsQO^l)^v, or ^ V 2 s v^ z 482. If the curve is the circumference of a circle, the centre of the circle may be assumed as the origin of coordinates. If then the axis of % is the intersection of the plane of the circle with the vertical plane, which is drawn perpendicular to it through the origin, and if E is the radius of the circle, and 2y = (y, the equation (25428) becomes 2E r 27? r 2ji r — J.sJ{2gIicoslcos2(p^2H) ~ J. ^\J{2gIicoslcos2(p-\-2JI) J ^sj {2 H -{-2 g R co&l — ^gRcosl sin^ g)) * If then II is greater than g Bcos\, which is similar to the case of § 470, let h^=2ff+2gBcosl^, ig R cos 1=^1^ sin^« ; — 256 — and tlic preceding equation becomes AVhen i is quite small, this equation admits the same reduction with that given (25l3i — 2523). If// is smaller than ^i? cos ^i, which is similar to the case of §469, let /^2 z=. 4:(/Bcosl sin^i . . sin qp sin I and the equation (25624), becomes, by the same reduction with that given in (2521^), V Vycosj^/ which when i is small gives for the time of oscillation of the simple pen- dulum in an oblique plane V 5'cos^j If// is just equal to^7?cos^^, the equation (25504) becomes ^ = V^(7^)l°Stan(J. + ic;). The case in which H differs but little from ^7? cos?,, may be subjected to the same treatment with that adopted in (2555_23). — 257 — MOTION OF A HEAVY BODY UPON A SIOVIN'G LINE. 483. If the heavy body moves upon a line, which has a motion of translation in space, the equation of motion becomes, by the form of argument and notation adopted in § 464, Z>^s=: — TFcos,', -l-^cos*. 484. If the motion of the line is uniformly accelerated and invariable in direction, the motion of the body upon the line is the same which it would be if the line were fixed, and the force a con- stant force which coincided in amount and direction with the re- sultant of ^ and — W. Thus if the line moves vertically downwards with an accelerated velocity, equal to that of a heavy falling body, the body moves upon the line with an uniform velocity. 485. If the line is straight, and if the motion of translation follows a law, dependent exclusively upon the time, so that if At denotes the law, by which the line moves in the direction of its length, the acceleration in the direction of the line is — TFcos4.= i>r A; and the value of s becomes in which a and h are arbitrary constants. The absolute motion of the point in any direction in space, as that of the axis of x^ , is repre- sented by the equation .Ti = (S — A) cos ^,^p COS l^, 33 — 258 — in which j9 denotes the perpendicular upon the line from the origin. If the line is vertical, and limited in its motion to the vertical plane of Xi ;?!, and if the axis of % is vertical, the equations which deter- mine the position of the point in space are When j) increases uniformly so that p is constant, these equa- tions give Xi =:zp t, Zi — a -\- —, Xi -\- — V2 ■^i > so that the path of the body in Space is a parabola, of which the axis is vertical. 486. If the line moves ivith an uniform motion in a straight line, the equation (2578) gives I>^tS = g cos l. The integral of the product if this equation multiplied by 2 D^s is {D,sf=f^2gcosiD,s = 2gJ^D,z=^2gz + a, in which a is an arbitrary constant. Hence if V denotes the velocity of the translation of the line, the square of the velocity of the point in space is (i>,.,)2=[v/(2^^ + «)-^cosrp + (Fsinrf ^2gz-\-a-{-V'- — 2Vcos]\l{2gz+a). The augmentation of the power of the moving body above its — 259 — initial power is, then, If the body iiad moved through the same path upon a fixed curve, the increase of power would have been ^=^(.--^J+^F^fcosr. If P is greater than Q, the excess of P above Q is the power acquired by the body from the accelerating motion of the line. But if Q exceeds P, the excess of Q above P is the power communi- cated by the body to the line, which involves the theory of many machines, of which heavy bodies are the moving forces. If, for ex- ample, the line moves horizontally, the power communicated by the weight is Q — P = V(v cos [— v° cos Ic). If, moreover, the initial velocity of the bod}^, relatively to the line, vanishes, the expression of the communicated power is re- duced to and when the direction of the line at its extremity coincides with that of its translation, this expression is still further reduced to 487. If the line is the circumference of a vertical circle, of which the radius is P, and if 9 is the angular distance of the body from the lowest point of the circumference, the equation of motion (2578) becomes R]y\{^z=L — TFcos^ — ^sin^. — 2G0 — When the motion of the line is in a vertical direction this equation becomes BI>}(f = — { W-\-ff) sin (f ; which, when (f is very small, is reduced to The integral of this equation is in which A and b may be determined by the equations 2sJ{ffB) DAogA = - TFsin 2 (^ |r+ b), 2^{ffB)I),b = WA[l—cos2{isJ§ + b)]', which give s/{ffB)D,{Asmb) = — AWsm(tJ^ + b)sin(tJQ = ^Aw[cos(2tJ^+b) — cosb] =^Aw[sm''ib—siii'(tJj^-\-H]; s/{c^B)I),{Acosb)^ — AWsm(t^^+b)cos{t^^) = —iAw[Hm(2i^l^+b) + smb]; when W is very small in comparison with y, A and B may be as- sumed to be constant in the first integration of the second members of these equations. When W is dependent upon the position of the body in such — 261 — a way, that, if '^ is a function of time, W(p = A TFsin (^ y/j -{-b) = ^; the preceding equations give V/(yi?)^sin5 = — J(ersin(/y/|)), s/{ffIi)Acosh = -f{^cos{tsJg). If, for example, 'tf= 2hsm?nt ; these integrals become "• m- R — g in which the arbitrary constants are determined so that A and h vanish with the time. 488. If the line rotates about the vertical axis of z, the equation of motion becomes, by the analysis and notation of § 467, jyfs = — iicos^a' -\-a^u cos" -\-(/ cos i = — wcos^a' -\-a^uDsU-\-(/ D,z. 489. When the rotation about the vertical axis is uniform, this equation becomes The integral of the product of this equation multiplied by — 262 — 2D,s is in which a is an arbitrary constant. 490. When the rotating line is straight and passes at a distance p from the axis, if s is counted from the foot of the perpendicidar {p) '^V^^ ^^^^ VmQ, the equation becomes (Btsy =: a^s^ sin^* -|- 2^s cos^ -f" ^V + ^ r= (a s sin ^ + ^ cot '.)+« + «y — (^ cot :) , of which the integral is easily found to be a^sin * = log(a^5sin^^4" 2^cos^ + 2a sin *Z>,5) + 3, in which h is an arbitrary constant. 491. The integral, in this case, can be just as readily obtained from the equation (26I29) which becomes a linear differential equa- tion. Its direct integral is or cos 2 . ats'mt , -j-, — atsmf. «''siir| ' in which A and B are arbitrary constants. This form is identical with that given by Vieille in his solution of the particular case of this problem, in which p vanishes. 492. If «<(^^cot:y— aV the value of s must be such as to render the second member of (2629) positive ; that is, the limiting values, between which the body cannot be contained, are defined by the equation «5osin^ = — ^cot^+i/[(^cot^)' — «V — «]. — 263 — The velocity of the body upon the line vanishes at these limits. If the initial direction of the motion of the body is towards these limits, it will approach them with a diminishing velocity ; and when it arrives at the nearest limit, the direction of motion w^ill be reversed, and it will thenceforth continue to move away from the limits. If « = — a- J? one of the limits is at the foot of the perpendicular (/>), and the other limit is above this foot, at the point for which 5 = ^cot" If « (^cos,'y— «y, the motion will always continue in the same direction along the line, {a -|- «y ) will express the square of the velocity of the body upon the line when it is at the foot of the perpendicular. The point of least velocity upon the line will be determined by the equation g cos I and the least velocity will be — 264 — 494. If « = (^^cot,^)'— «V the direction of the motion along the line is not subject to reversal ; for, in this case, the equation (2629) becomes DtS :=. a s sin I -\-- cot '^ ; of which the integral is ((jl s Sin ^ \ ^ cos J ' / The time of reaching the point, at which g cos I a^sin^j' that is, the point, at which the velocity vanishes, becomes infinite ; or in other words, the body never reaches this point, at which its direction of motion is to be reversed ; or if the body is placed at this point without any initial velocity along the line, it will remain sta- tionary upon the line. 495. If the 7'otating line is the circumference of a circle, of which the radius is R, let the origin be assumed so that the centre of the circle may be upon a level with the foot of the perpendicular (jo), let fall from the origin upon the plane of the circle. Let then h denote the distance of the centre of the circle from the foot of the perpendicular, 9) the angular distance upon the circumference of the body from the lowest point of the circumference, and the values of z and ii, in equation (2622), arc given by the — 265 — equation 0r= li COS (f sin^-|-^cos^, u^ z={/c-{- li sin (pf-\- {p sin I — 7? cos (p cos ^f whence equation (2622) becomes -\-2{ff — a^p cos^) it sin^ sin cp — «^i?^sin^^ cos^ (p . The points of maximum and minimum velocity along the arc are, therefore, determined by the equation a^kM cos (pi — ((/ — a^p cos ^) i?sin^ siu^j -|- «^i?^sin^^ sin ^i cos 9)1 = 0, and are, consequently, at the intersections of the circumference with the equilateral hyperbola, which is described in the plane and passes through the centre of the circle, of which one of the asymptotes is horizontal, and the polar coordinates (r2, (p^) of the centre, with reference to the centre of the circle, are given by the equations, ^2 sin ^2 = — ^'^ cosec^ I , ^2 cos ^2 = ^ cosec^ — 2^ cotf . This hyperbola cannot cut the circumference in less than two points ; and there are four points of intersection when the distance from the centre of the circle to the nearest point of the branch of the hyperbola, which does not pass through it, is less than the radius of the circle. The polar coordinates [r^, ^3) of this nearest point of the second branch of the hyperbola are given by the equations tan (p3=\J tan (p2, 34 rg ■= r^ cos (po, sec^ (^3 . — 266 — 496. When the body is originally placed at one of the points of maximum or minimum velocity, without any initial velocity along the circle, it remains stationary upon the curve ; but its position upon the curve is one of stable equilibrium, when it is placed at a point of maximum velocity, and a position of unstable equilibrium, when it is placed at a point of minimum velocity. When the body is originally placed upon the curve, without any initial velocity along the line, at a point different from these points of maximum or minimum velocity, it oscillates about that point of greatest velocity from which it is not separated by a point of least velocity ; its oscillations embrace both the points of great- est velocity, when the velocity is sufficient to carry it through either of the points of least velocity, that is, when the velocity, which corresponds to the initial point in the general equation (265;), is less than that which corresponds to one of the points of least velocity. When the initial velocity of the body is greater than the excess, which is given by equation (265;) of the velocity at the initial point above the least of the minimum velocities, the body constantly moves, in the same direction, through the entire circumference. 497. The case in which the initial velocity of the body is just equal to the excess, which is given by equation (2667) ^^ the velocity at the initial point above either of the minimum veloc- ities, admits of integration. In this case, it is easy to express the equation (2667) in the form R [Dt cj))^ = 2 a^ k (sin c/) — sin f/ij) — a^R m.v?l (cos^ (p — cos^ (pi) -[-2 ([/ — a^p 0,0^1) sin^(coscj) — cosg)i), which by means of (2652o) assumes the form R{D^(.pf=za^i^iv?l [2^2 cos (9 -{-^>2) — 2 /a cos (9^1 -f- 92) — R cos^ ip -\- R cos^ 9)1] . — 267 — The condition for the determination of the point of min- imum velocity gives also the equation 2 ^2 sin {(pi -\- (f>2) = ^ sin 2 (pi , which substituted in the previous equation with the notation ^=^(p — (pi) H sin ('jTi — T'2) sin (gii + qpjj) gives {D, + (pi) — //] . If, therefore, H is negative and absolutely greater than unity, that is, if (p^i is not in the same quadrant with (p2, the value of fp is unlimited ; but if // is less than unity, the limits of are given by the equation cos2(^4-f/)i)— //. The integral of the equation (267ii) is at sml^(icos2(pi—^ II) 1 sin ( + f/i) v/(cos2(jri — B) — sin qri v/Ccos 2 (d) - {- (fi) — H '] o cos (a> + S'iC0sy, D,s =^ — S. 507. The integral of the latter of these equations is Let S have the form S=a-\-hs' -\-es'\ hi which a and e are positive, in the case of nature, and h + \l{iae)>0, because S is always positive when / is positive. The correspond- ing integral of (27I17) is in which A is an arbitrary constant, and the former integral cor- — 272 — responds to the case of P<^Aae,whi\e the latter corresponds to b^^Aae. The velocity vanishes after the time ^o given by the equation 4 = ^- .;.. '_.-. tan^-^^; These values are infinite in form, when P =:z Aae; but, in this case, the integral is , A _\ ^^ /d_J_ ^ ^ — ^ "I b{bs'-\-2a) — ^ I 2 77+ * ' so that the velocity vanishes, when \J{ae)' These values become infinite in form when both h and e vanish, but, in this case, which includes that of friction upon a straight path, the integral is a a and the instant, at which the velocity vanishes is determined by the equation When a vanishes, the value of t(^ is actually infinite, so that the velocity of the body can never be wholly destroyed by any such form of resistance. It would seem, from the preceding equa- — 273 — tions, that the direction of motion would be reversed after the time (4). But this conclusion, which is absurd, because it would give a resistance the power of creating motion, arises from the defective forms of notation w^hich do not express the solution of continuity corresponding to the abrupt ceasing of the friction at the instant of the suspension of motion. 508. When the resistance is simply that of friction arising from the pressure of tJie moving body upon the line, to ivhich its motion is restricted, let p denote the direction of the perpendicular to the fixed line, which is drawn in the common plane of the direction of the external force and of that of the line, dv the elementary angle made by two successive radii of cur- vature to the fixed line, and a the coefficient of friction, and the equation of motion becomes by (245i8) 509. When there is no external force, this equation becomes Dts' ==: — as' v' ', the integral of which is 102:/==^ — av . 'o in which A is an arbitrary constant. Another integration gives C av — A C(j^ av — A\ C( av—A\ in which c is the Naperian base, and q the radius of curvature of the fixed line. 35 — 274 — 510. If the fixed Hue is the involute of the circle, and if its equation is the equation (273o8) becomes in which B is an arbitrary constant. 511. If the fixed line is the logarithmic spiral, and if its equa- tion is i)=: Re , the equation (27328) becomes a-\-b ' ' in which B is an arbitrary constant. 512. If the fixed line is the cycloid, and if its equation is Q = 4 7? sin I' , the equation (27328) becomes I = .,. [a sni V — cos v) c + -" in which B is an arbitrary constant. 513. When the resistance of the line is constant, and the resisting medium is moving ivith an uniform velocity in an invariable direction, and the resistance arising from it is proportional to the velocity in the medium, let a be the constant resistance of the hne, h the resistance of the medium for the unit of velocity, and I) the velocity of the medium, — 275 — and if the direction of the motion of the medium is assumed for that of the axis of x, the equation of motion becomes Z>,/ = Z>j, 12 — a — lis'x cos \ in which it is carefully to be observed that the sign of a must be reversed simultaneously with the direction of motion. 514. When the fixed line is straight and there is no external force the integral of the equation (2755) becomes log(/ — 5 cos ^ -|-^j =: ^ — ht in which A is an arbitrary constant. When a<^hh cos \ , the velocity of the body will never be destroyed, but will constantly approximate to cos^ — J. But when a'^ bh cosl, the velocity will vanish after the time t^, determined by the equation log (^ — b cos i):=A — hto. If the initial velocity of the body had been negative, the equation of motion would have assumed the form log(— /+^cos^ + ^^) = — yl + /^/; so that the velocity would have vanished after the time 4, deter- — 276 — mined by the equation The body would then have remained at rest unless the con- dition (275i4) had been satisfied, in which case its subsequent motion would be defined by the equation (275ii). 515. When a heavy hody moves upon a fixed straight line, and the resistances consist of a ^constant resistance, arising from the friction along the line, and also of a resistance arising from a resisting medium, which has a uniform motion in the direction of the fixed line ; and ivhen the re- sistance of the medium is proportional to the square of the velocity of the body in the medium, let a be the constant of friction, h the velocity of the medium, and h the resistance of the medium for the unit of velocity. The line may be assumed to be vertical without diminishing the generality of the investigation and the equation of motion will be D,s' =g — a — h{s' — h)\ in which the signs of a and h must be reversed simultaneously with those of / and [s' — h) resj)ectively. The equation of motion has precisely the same form with that of § 507, so that the forms of the integral are the same which are there given, but the constants are not subject to the restrictions of that section. If, then, the initial velocity is upward and exceeds that of the medium, when the medium is also moving upwards, the ascend- ing velocity decreases by the law expressed in the equation /_} = ^?^tan[(<-hP, the body is brought to a state of rest, in which it continues per- manently if g — a hb'^, the body moves from the state of rest with an increasing descending velocity of which the law is expressed by the equation in which t^ must be determined so that the instant of rest coincides — 278 — with that given by the equation (2778). The increasing velocity continually approximates to that which is determined by the equation g^a = h{s —hf. Tlie state of rest to which the body is brought, when the medium is not moving upwards, is permanent if a — g'^hh'^. But if, on the contrary, a — g "^hb"^ the body moves from the state of rest with an increasing descending velocity, of which the law is expressed by the equation s' _ } = y/ ?=i^ tan [(i; - r,) y/ (A (^ - «) )] , when in which Ti must be determined so that the instant of rest coincides with that given by the equation (27630). This law of motion con- tinues until the instant Tj, when the downward velocity of the body becomes the same with that of the medium; and after this instant, the law of increasing velocity of descent is expressed by the equation (27729) ; so that the velocity continually approximates to that which is determined by the equation (278^). But when the body begins to descend from the state of rest, and — 279 — the law of descent is expressed by the equation SO that the increasing velocity constantly approximates to that which is determined by the equation a—g = h(s—hf. If the initial velocity is downward, and exceeds that deter- mined by the equation (2784), the decreasing velocity when is expressed by the equation s'-b = ^i^Goi\_{t-r)sj(h(g-a))-], in which t is an arbitrary constant. If, therefore, the motion of the medium is downward, or if it is upward and the condition (27724) is satisfied, the decreasing velocity continually approximates to that which is determined by the equation (27729). But if the motion of the medium is upward and the condition (2772i) is satisfied, the body is brought to a state of rest which is permanent if the condition (277];) is also satisfied. If, however, the condition (277ii) is satisfied by the upward motion of the medium, the body leaves the state of rest and ascends with an increasing velocity, which is defined by the equation s'-i = y'-^'Cot[(<-r,)v/(A(y + «))], in which Tj must be determined so that the instant of rest coin- cides with that which is given by the equation (279i5). The — 280 — ascending velocity continually approximates to that which is determined by the equation (277i5). If the initial velocity is downward, and exceeds that of the medium, when the medium is also moving downwards, the de- scending velocity, when decreases by the law, expressed in the equation in which t is an arbitrary constant. This law of descent continues until the body is brought to rest, when the medium is not moving downwards; but when the medium is moving downwards, the law continues until the instant t, when the velocity of the body is the same with that of the medium. After this instant, the law of decreasing velocity becomes ^'-b = ^''-^T<^n [(t: - V (/* (« -9))] , which continues until the body is brought to rest, when the condi- tion (2789) is satisfied. But when, on the contrary, the condition (27812) is satisfied, the body continues to move forever with the law of decreasing velocity expressed in (28O19), and the velocity continually approximates to that, which is determined by the equation (2797). When the body has been brought to the state of rest, the condition and laws of leaving it are the same with those defined in (27923_3j), when — 281 THE SIMPLE PENDULUM IN A RESISTING MEDIUM. 516. When the curve is the circwnfe?'enee of a vertical circle, the problem is that of the simple pendulum in a resisting medium. If the arc of vibration is supposed to be so small that its powers, tvhich are higher than the square mag be neglected, and if the 7'csistance arising from the medium is supposed to be proportional to the velocity, and to he combined zvith a constant friction, let a be the friction, and h the resistance of the medium for the unit of velocity, and the equation of motion becomes, by adopting the notation of § 487, in which the sign which precedes a, must be the reverse of that of Dt (p . The integral of this equation is in which I -^« I (To — h^^ ' 7J k = ^ j^cosa ^ h = \/ ^sina. and the arbitrary constants have been determined so that the initial angular velocity (9J,) shall be the maximum velocity, and, therefore, the initial value of y is I S a — 9 36 — 282 — 517. The equation (28I21) only applies to the first vibration and for the {m-\- 1)'' vibration, the correct equation is 9 = ±^ + fc-^^^'-'-hmkit-T„), in which t,„ is the instant of the maximum angular velocity (ff^ of that vibration and the doubtful sign is alternately positive and negative for the successive oscillations, so that the position of maximum velocity is always upon the descending portion of the oscillation. 518. The angular velocity of vibration is expressed by the equation / / —^h{t — T„.) cos [k (t — T„.) -f «] r — 9m c COS a ' and it vanishes for the instants which correspond to the beginning and end of the oscillation. The whole time of oscillation is, therefore, T=-T^7i i - sec ct, k \ (/ which is invariable, although it exceeds the time of vibration in a vacuum, in consequence of the factor, sec a . 519. The angular deviations of the pendulum from the verti- cal at the beginning and end of the oscillation are given by the equation — Ra — , I R (a-\-\n) tan a — 283 — whence the whole arc of the {m -\- 1)'' vibration is ■w ^ f IR « tan a ^ /i , \ v^^l^>^d -c Cos (1 71 tan «). 520. The angular deviations of the pendulum from the ver- tical at the end of one vibration and the beginning of the next are identical, but the deviation from the point of maximum velocity is, on account of the change in the position of this point, diminished by the quantity 2Ra 9 The successive values of the maximum velocity are therefore connected by the equation or f , — 71 tan a ^ / ^ — (a-\-\Ti) tan a (pm + i = fpmC _2«y/-c V -r. y The general expression for the maximum velocity is then found to be /■ , — 7n7Ztana o /^ — («4- i;r) tan « /c — '"7r tana — 1\ which, on account of the smallness of a and a, may be reduced to / /■ — m 7Z tan « r> / R fpm = ^>oO — 2maU -. The corresponding value of the arc of vibration is E ^ \/ [2 Ac — ^^ /J' -{-2 g Ji cos lA. cos ((p-\- ia)^' — 285 — The signs which precede the quantities h and ^i must be re- versed in the alternate oscillations. 522. The angle of greatest deviation from the vertical for the (m -f- l)st oscillation is determined by the equation — (fm tan u . V c ^ ^cos(9)^ — /^,) g a COS jt* If J is adopted as the sign of finite differences, this equation gives, when fi is so small that its square may be neglected, J [cos (p„, — (sin 9,„ — (p^ cos (p^) fi'] = 2 (sin (p^ — (p^ cos (p„,) ^ . When the oscillations of the pendulum are so small that the fourth power of y,„ may be neglected, and also the product of jit by fpm ^ ^my this cquatiou is reduced to of which the approximate integral is qPo 523. The substitution of (2855) reduces (28427) to the form E {D.cpf = cos {cp + .a) - c- ^'^ + ^'^"^ '"" ^ cos (9,, - ^) , 2 g cos [I which, when in is so small that its square may be neglected, becomes 7? — (A 9)2= cos {(p+fi) — COS (9),, — ^) + COS (p„, {(p + (p,n) u. = COS 9) — COS (p„, — in [sin 9 + ^^^ ^m — (^ + ^m) cos f/)„,] . When the oscillations are very small, this equation may be — 286 — still further reduced to which gives The integral of this equation is )i3:: — cos(^'i/|4-fj) — sin/Jcosnogtan(i:T+ ^^^^+^ ), ^d{A cos b)=^ — sin (/y/| + /i) + sin/^sini log tan(i n + ^-^^|i-^) ; whence d { = - ^ cos (y I + i + (i) - 1^ sin /i log tan (i ^ + (vi+_*) . The changes of A and b in a vibration are J b:= -T-sm(i log tan i /i? . The combination of these values give ^^^_^_;r^'-cos/^ — -V---^'cosf^, J b=z~sm(-i lopj tan i /:? — n - sin /•> — -U - ^ sin f:/ . The change of ^ is exhibited in the motion of the pendulum — 291 — by a change in the time of vibration, which differs from that Avhich it would be in a vacuum. The diflerence is y 9 n 526. The vibration of the pendulum may be regarded as affected by the medium not only in consequence of its direct action as resistance, but also indirectly, because a portion of the medium may be regarded as composing a part of the moving body, and its motion is sustained by the action of gravitation upon the body. If, then, q denotes the ratio of the mass of that portion of the medium which moves with the body to the mass of the body, the motion of q may be assumed to have a period identical with that of the body, and an amplitude of excursion proportional to that of the body, so that its velocity may be of the form r = A' J y/l cos (y ! + «-/}'). The resistance, then, arising from the preservation of the motion of q may be expressed in 9° by the form The similarity of this form to that of (2884) shows that the corresponding influence upon A and h may be expressed by the equations z/log^r=z — l^sin/f, — 292 — The importance of this form of resistance was first noticed by DuBUAT and has been investigated experimentally by Dubuat, Bessel, and Baily. The formulae (29O27) and (29I29) may be adopted as a guide in the conduct of these and similar investigations. 527. In the application of the preceding formuloB to the re- duction of experiments, the quantities a^ h, Jc, and q are inversely proportional to the density of the body, and directly proportional to the density of the medium, and for bodies of similar forms they are nearly in an inverse ratio to their linear dimensions. For pendulums of different lengths, k is proportional to the length of the pendulum, and h to the time of vibration. If H^ denotes the resistance of the medium which is proportional to the velocity for the unit of weight and the unit of surface, and if ^ denotes the resistance which is proportional to the square of the velocity for the same unit of weight and surface, the values of h and k, for the units of weight and surface, are /; — — TTT k = \gRII^, 528. The best experiments which have been made with the pendulum are almost wholly free from any constant term of resist- ance, so that, in their discussion, this term may be neglected which reduces the formula (290.26) to the form AA = — \ TH^Aco^(i — ^ RH^A^co^(i, of which the approximate integral is 1°= (1 + 8-TO:) - 1«S (1 + s-^) = J '» TII, cos t. ■ 529. In order to illustrate these formulas, they may be ap- plied to some of the experiments which have been actually made, — 293 — and in which the diminution of the arc of vibration has been ob- served. For this purpose the observations of Newton, Dubuat, BoRDA, Bessel, and Baily are selected, and the formula (29228) is found to be applicable to all these experiments, although the values of Hi and H2 are different for the different experiments. The unit of length which is here adopted is the meter, the unit of weight is the chiliogramme, and that of time is the mean solar second. The measures and weights are, however, given in the form in which they were actually observed. 530. In Newton's first series of experiments upon the dimi- nution of the oscillations of a pendulum, a wooden sphere of 61 English inches in diameter, weighing 57 2^? ounces, of about 0.56 specific gravity, and susjDcnded by a fine wire so as to give 10 i feet for the length of the pendulum, was vibrated until the arc of descent was diminished one fourth or one eighth of its initial extent, and the number of vibrations was recorded. From the re- duction of these observations, I have obtained for the values of 11^ and II2 ^1=0.0223 sec/?, 7/2 =0.4473 sec/?. In Newton's second series of experiments, a leaden sphere of 2 inches in diameter, weighing 261 pounds, and suspended so as to give 10? feet for the length of the pendulum, was vibrated in the same way as in the former series. From the reduction of these observations, I have obtained ^1= 0.2044 sec /?, ^2 = 0.701 sec/?. To test the accuracy of these reductions, and their conformity with the given observations, I have computed the lengths of the 294 observed arcs of vibration, and have placed them in the following table for comparison. COMPARISON OF NEWTON S EXPERIMENTS UPON VIBRATIONS OF THE PENDULUM WITH COMPUTATION. WOODEN SPHERE. LEADEN SPUERE. m Computed Observed C m Computed An Observed C in. in. in. in. in. in. 64.08 64 .08 64.03 64 .03 H 56.02 bQ> .02 30 56.04 56 .04 22f 47.91 48 —.09 70 47.92 48 —.08 31.86 32 —.14 31.92 32 —.08 181 27.92 28 —.08 53 28.00 28 0. 41f 24.19 24 .19 121 24.07 24 .07 15.99 16 —.01 16.01 16 .01 35i 14.01 14 .01 901 13.99 14 —.01 83i 11.99 12 —.01 204 11.99 12 —.01 8.04 8 .04 8.05 8 .05 69 7.01 7 .01 140 7.01 7 .01 lG2i 5.95 6 —.05 318 5.95 6 —.05 4.01 4 .01 4.03 4 .03 121 3.50 U 0. 193 3.49 H —.01 272 2.99 3" —.01 420 2.97 3 —.03 1.98 2 —.02 2.04 2 .04 1G4 1.74 H —.01 228 1.74 If —.01 374 1.52 H .02 518 1.46 l| —.04 1.00 1 0. 226 .88 1 0. 510 .75 1 0. With these values of Hi and II2, a minute arc of vibration of the wooden sphere would be reduced one eighth part in 446 vibrations, and one fourth part in 961 vibrations, and a minute arc of vibration of the leaden sphere would be reduced one eighth part in 290 vibrations, and one fourth part in 625 vibrations. 531. DuBUAT vibrated in water a sphere of 2.645 French inches in diameter, weighing in air 40068 grains, and in water 36448 grains, and suspended so that the length of the pendulum — 295 — was 36.714 inches; he observed the arc of descent at each succes- sive oscillation. From these observations, I have obtained a result which corresponds with his own in respect to the law of diminution of oscillation, and which gives for the values of II^ and 11^ in water ^2 =378.7 sec/?' DuBUAT also vibrated in air a paper sphere of 4.0416 inches in diameter, weighing in air 155 grains, with a density 11.33 times as great as that of air, and suspended by a fine thread so that the length of the pendulum was 36.714 inches. From these observa- tions, I have deduced ^2 = 0.37 sec /i. The following table contains the comparison of Dubuat's experi- ments with the computations derived from the values of //i and H^, COMPARISON OF DUBrAX's EXPERIMEXTS UPON THE DIMINUTION OF THE ARC OF VIBRATION OF A PENDULUM WITH COMPUTATION. SPHERE IN WATER. SPHERE IN AIR. m Computed Observed An 1 C m Computed A. Observed C in. in. in- in. in. in. 12.00 12.00 0. 11.90 12.00 —.10 1 9.21 9.25 —.04 1 10.10 lO.dO .10 2 7.47 7.42 .05 2 8.77 8.70 .07 3 6.28 6.25 .03 3 7.75 7.79 —.04 4 5.42 5.33 .09 4 6.94 6.96 —.02 5 4.77 4.75 .02 6 4.25 4.25 0. 7 3.84 3.83 .01 8 3.50 3.48 .02 9 3.22 3.23 —.01 10 2.97 2.98 —.01 _ 296 — 532. BoRDA vibrated a platinum sphere of 16 J lines in diameter, weighing with the wire and screw 9963 grains, and suspended by a wire so that the length of the pendulum was 3.95497 metres. These observations give for the values of Hi and 11^ in air ^1=: 0.10722 sec/:?, ^,r= 0.6267 sec/:?. In his observations for determining the length of the seconds pendulum, this same pendulum was vibrated by Borda, and the lengths of its arcs of vibration were observed. From the mean of these observations, I have obtained the values of H^ and II^, //i= 0.11214 sec /i, ^2 =0.6564 sec /i. Borda vibrated the same sphere with a smaller wire, so that the weight was reduced to 9958 grains, and the length increased to 3.95597 metres. From these observations I have derived ^1 = 0.1134 sec /^, 7/2 = 0.590 sec /^ The comparison of Borda's experiments with the computations based upon these values of H^ and H^ is contained in the following tables. COMPARISON OF BORDA's OBSERVATIONS UPON THE DIMINISHED VIBRATIONS OF THE PENDULUM WITH COMPUTATION. First Experiment icith direct re/cnnce to the Dixiinution of the Arc of Vibration. m Computed Observed An C m Computed Observed An C 120'0 120'0 0. 12600 4.2 4.1 1 0.1 1800 61.2 61.2 0. 14400 2.8 2.7 3 GOO 35.6 35.4 .2 1 6200 1.9 1.8 5400 22.1 21.9 .2 18(100 1.3 1.2 7200 14.2 14.1 .1 19800 0.9 0.8 9000 9.4 9.4 0. 2 1 600 0.6 0.5 10800 6.2 6.3 —.1 36000 0.002 Very minute. — 297 — Experiments for determining the Length of the Second's Pendtihtm with the Pendulum used in the First Experiment. m Mean Value. Computed An Observed ^,n a—0 Computed Observed An C—0 Computed Observed C—0 d 64 1 67 m 63' 63' / 2169 321 32 1 2 34 34 32 32 4338 18' 19 — 1 181 19 \ 2 18 18 6507 lOi 11 1 11 11 101 11 2 8676 6l 7 -\ 61 7 -i 6 6 60 60 61 61 64^ 641 2169 31 31 31^ 311 321- 321 4338 171 17 1 171 18 i 2 18 171 2 6507 10 10 10 10 10^ 10 i 8676 6 6 6| 6 1 2 63 63 68 68 61 61 2169 32 32 34 341 1 — 2 31 ^ 311 4338 18 18 19 191 1 — 2 171 17 \ 6507 lOi 10 i 11 lU _L 10 10 8676 6 6i J. 4 61 7 1. 6 6 591 591 571 571 62 62 2169 301 301 30' 30 3U- 311 4338 17" 17' 17 17 18 17 1 6507 10 10 91 10 -\ 10^ 10 JL 2 8676 6 6 6 6 67 67 65 65 63 63 2169 34 34 33 331 — i 32 32 4338 181 19 1. 181 i8i 18 171 2 6507 11 11 101 11' —h 101 10 J. 2 8676 61 6 i 61 ^\ 6 6 71 71 591 591 2169 35 341 2 31 31 4338 191 19 JL 2 171 17 h 6507 11 11 10 10 8676 7 7 6 6 Experiments for determining the Length of the Second's Pendulum with the Second Pendulum. m Comp'd An Observ'd C-0 m Comp'd ( Jbserv'd A,n G-0 m Computed An Observed An G-0 551 54 79 / 79 111 110 1 1 1575 341 35 I 1538 47 47 1445 641 641 3150 221 23 2 3114 30 30 2970 401 40 2 4725 15 16 —1 4690 191 20 4495 26 26 6300 10 101 1 2 6266 13" 14 —1 6020 171 18 1 2 7875 7 H L 2 7842 9 H L 7545 12" 12 9450 1 5 5 9418 5 61 i 2 9070 8 81 1 38 — 298 — 533. In Bessel's experiments made for the determination of the length of the second's pendidum of Konigsberg, a brass sphere of 24.164 hnes in diameter, weighing 0^695364 was suspended so that the length of the pendulum was 1305.3 lines. From his ob- servations with this pendulum, I have found these values of Hi and /^. //i = 0.05698 sec /^, ^^2= 0.529 sec /':J. The same sphere was also vibrated with a length of pendulum of 441.8 lines, from the observations of which I have deduced //i = 0.0452 sec f:f, ^2 =0.587 sec/":?. Bessel also vibrated an ivory sphere, weighing 0M5112, and having a diameter of 24.094 lines, with each of the preceding lengths of pendulum. From his observations with this sphere and the long pendulum, I have obtained //i = 0.05517 sec fi, 7/2 = 0.512 sec f:?; and from his observations w4th the short pendulum, ^1 ==0.0509 sec /^, 7/2 = 0.282 sec ^^. In Bessel's experiments for the determination of the length of the second's pendulum at Berlin, a hollow cylinder was vibrated, of which the diameter of the base was 15.305 lines, and the altitude _ 299 — 15.296 lines, weighing, with its appendages, Avhen it was filled with lead, 0^67920, and when it was empty, 0^22595. It w\ns suspended in two different modes, in one of which the length of the pen- dulum w^as 1304.8 lines, when the cylinder was filled, and 1303.8 lines, when it was empty ; and, in the other mode of suspension the length was 440.9 lines when the cylinder was filled, and 440.7 lines when it w^as empty. From his observations with this pendulum, I have obtained the following values of H^ and i^. When the cylinder was full, and the suspension was long, the values were ^i:= 0.08544 sec /^, H^ = 0.733 sec (i ; when it was full, and the suspension short, they were ^i=zi 0.07026 sec fi, ^2 =.0.724 sec /^. When the cylinder was empty, and the suspension long, the values were ^=0.09578 sec /'i, //=: 0.559 sec (S ; when it was empty, and the suspension short, they w^ere ^1= 0.07003 sec /i, ^2 =0.270 sec/':?. In order to compare the theory of these values w^itli ex- periment, all the values of observation have been recomputed, and the comparisons are contained in the following tables. — 300 — COMPARISON OF BESSEL's OBSERVED ARCS OF VIBRATION OF THE PENDULUM WITH THE COMPUTED ARCS. 1. Experiments with the Brass Sphere and Long Suspension. m Computed Observed An C Computed -4m Observed C Computed Observed Am C 38.3 38.3 39.0 39.0 39.5 39.5 500 33.7 33.8 —.1 34.2 34.2 34.6 34.6 1000 29.7 29.8 —.1 30.2 30.1 .1 30.5 30.5 1500 26.4 26.4 26.8 26.8 27.1 26.8 .3 2000 23.5 23.6 —.1 23.9 23.8 .1 24.1 23.9 .2 2500 21.0 20.9 .1 21.3 21.3 21.6 21.6 3000 18.8 18.8 19.1 19.2 —.1 19.3 19.3 3500 16.9 16.9 17.2 17.2 17.4 17.3 —.1 4000 15.3 15.4 —.1 15.5 15.5 15.7 15.7 39.7 39.9 —.2 39.0 39.3 —.3 39.6 39.7 —.1 500 34.8 34.6 .2 34.2 34.1 .1 34.7 34.8 —.1 1000 30.7 30.4 .3 30.2 30.0 .2 30.6 30.5 .1 1500 27.2 27.1 .1 26.8 26.4 .4 27.1 26.9 .2 2000 24.2 24.1 .1 23.9 23.5 .4 24.2 24.0 .2 2500 21.6 21.5 .1 21.3 20.9 .4 21.6 21.4 .2 3000 19.4 19.3 .1 19.1 18.5 .6 19.4 19.3 .1 3500 17.4 17.3 .1 17.2 16.4 .8 17.4 17.3 .1 4000 15.7 15.5 .2 15.5 14.6 .9 15.7 15.5 .2 38.6 38.6 40.0 40.3 —.3 40.1 39.9 .2 500 33.9 33.9 35.1 34.9 .2 35.1 35.2 1 1000 29.9 29.9 30.9 30.8 .1 31.0 31.0 1500 26.5 26.6 —.1 27.4 27.2 .2 27.4 27.5 1 2000 23.7 23.6 .1 24.4 24.2 .2 24.4 244 2500 21.1 21.2 —.1 21.8 21.8 21.8 21.9 — 1 3000 19.0 19.0 19.5 19.5 19.6 19.6 3500 17.1 17.1 17.5 17.4 .1 17.6 17.6 4000 15.4 15.4 15.8 15.6 .2 15.8 15.9 — 1 39.1 39.1 39.3 39.2 .1 38.8 38.5 .3 500 34.3 34.3 34.5 34.5 34.0 34.0 1000 30.3 30.3 30.4 30.5 —.1 30.1 30.2 — 1 1500 26.9 26.9 27.0 27.1 —.1 26.7 27.0 —.3 2000 23.9 23.9 24.0 24.2 .2 23.8 24.0 —.2 2500 21.4 21.4 21.5 21.8 —.3 21.3 21.5 —.2 3000 19.2 19.3 —.1 19.3 19.4 —.1 19.1 19.3 —.2 3500 17.2 17.3 —.1 17.3 17.5 —.2 17.2 17.4 —.1 4000 15.5 15.6 —.1 15.6 15.7 —.1 15.5 15.5 — 301 1. Expe\ •iments wi th the Brass Sphere and Long Susj)ension. — Continued. m Computed A. Observed C Computed An Observed C—0 Computed An Observed An C 39.1 39.1 37.8 37.7 .1 39.6 39.7 —.1 500 34.3 34.2 .1 33.2 33.3 —.1 34.7 34.7 1000 30.3 30.2 .1 29.4 29.3 .1 30.6 30.6 1500 26.9 27.0 —.1 26.1 26.2 —.1 27.1 27.1 2000 23.9 24.0 —.1 23.2 23.4 —.2 24.2 24.1 .1 2500 21.4 21.5 —.1 20.8 20.9 —.1 21.6 21.6 3000 19.2 19.2 18.7 18.7 19.4 19.4 3500 17.2 17.3 —.1 16.8 16.7 .1 17.4 17.3 .1 4000 15.5 15.5 15.1 15.2 —.1 15.6 15.6 .1 39.0 39.0 41.7 41.6 .1 39.4 39.4 500 34.2 34.1 .1 36.5 36.6 —.1 34.6 34.5 .1 1000 30.2 30.1 .1 32.1 32.3 .2 30.5 30.4 .1 1500 26.8 26.5 .3 28.4 28.6 —.2 27.0 27.1 — .1 2000 23.9 24.0 —.1 25.3 25.5 .2 24.1 24.1 2500 21.3 21.4 —.1 22.5 22.7 .2 21.5 21.6 —.1 3000 19.1 19.2 —.1 20.2 20.3 —.1 19.3 19.4 —.1 3500 17.2 17.2 18.1 18.1 17.3 17.3 4000 15.5 15.5 16.3 16.3 15.6 15.6 39.2 39.4 —.2 38.6 38.6 38.5 39.3 —.8 500 34.4 34.4 33.9 33.4 .5 33.8 34.2 —.4 1000 30.3 30.3 29.9 29.9 29.8 30.0 —.2 1500 26.9 26.9 26.5 26.7 2 26.5 26.3 .2 2000 24.0 24.0 23.7 23.6 .1 23.6 23.2 .4 2500 21.4 21.5 — 1 21.1 21.4 —.3 21.1 20.5 .6 3000 19.2 19.2 19.0 19.2 2 18.9 18.3 .6 3500 17.2 17.1 .1 17.1 17.2 — 1 17.0 16.3 .7 4000 15.6 15.4 .2 15.4 15.4 15.3 14.5 .8 40.0 39.9 .1 39.9 39.6 .3 39.3 39.0 .3 500 35.1 34.9 .2 35.0 35.0 34.5 34.5 1000 30.9 30.9 30.8 30.9 —.1 30.4 30.5 — 1 1500 27.4 27.5 — 1 27.3 27.5 2 27.0 27.1 — 1 2000 24.4 24.4 24.3 24.3 24.0 24.1 — 1 2500 21.8 21.9 — 1 21.7 21.7 21.5 21.5 3000 19.5 19.7 — 2 19.5 19.5 19.3 19.3 3500 17.5 17.6 —.1 17.5 17.3 .2 17.3 17.3 4000 15.8 15.8 15.8 15.5 .3 15.6 15.5 .1 39.7 39.8 —.1 38.9 38.8 .1 38.7 38.7 500 34.8 34.8 34.1 34.0 .1 34.0 34.3 —.3 1000 30.7 30.7 30.1 30.2 — 1 30.0 29.9 .1 1500 27.2 27.2 26.7 26.9 2 26.6 26.4 .2 2000 24.2 24.2 23.8 24.0 —.2 23.7 23.5 .2 2500 21.7 21.8 —.1 21.3 21.4 —.1 21.2 21.1 .1 3000 19.4 19.4 19.1 19.2 —.1 19.0 18.9 .1 3500 17.4 17.4 17.2 17.2 17.1 16.8 .3 4000 15.7 15.6 .1 15.5 15.4 .1 15.4 15.2 .2 302 1. Experiments with the Brass Sphere and Long Suspension. — Continued. m Computed ObserveJ C—0 Computed An Observed C—0 Computed An Observed C—0 38.7 38.7 39.3 39.3 39.1 39.2 —.1 500 34.0 34.1 —.1 34.5 34.7 —.2 34.3 34.2 .1 1000 30.0 30.0 30.4 30.2 .2 30.3 30.3 1500 2G.6 26.6 27.0 27.0 26.9 27.0 —.1 2000 23.7 23.6 .1 24.0 24.1 —.1 23.9 23.9 2500 21.2 21.2 21.5 21.5 21.4 21.4 3000 19.0 19.0 19.3 19.3 19.2 19.3 —.1 3500 17.1 16.9 .2 17.3 17.3 17.2 17.2 4000 15.4 15.3 .1 15.6 15.5 .1 15.5 15.5 39.0 39.0 39.8 39.7 .1 500 34.2 34.1 .1 34.9 34.9 1000 30.2 30.1 .1 30.8 30.8 1500 26.8 26.8 27.3 27.2 .1 2000 23.9 23.7 .2 24.3 24.3 2500 21.3 21.2 21.7 21.7 3000 19.1 19.2 —.1 19.4 19.4 3500 17.2 17.2 17.5 17.4 .1 4000 15.5 15.4 .1 15.7 15.6 .1 2. Experiments with the Brass Sphere and the Short Suspension m Comimted An Observed An C Computed An Observed C—0 Computed An Observed Am C 14.4 14.65 —.2 13.2 13.5 —.3 12.4 12.4 560 13.5 13.7 —.2 12.4 12.7 —.3 11.7 11.6 .1 1120 12.7 12.8 —.1 11.7 11.9 —.2 11.0 10.9 .1 1680 12.0 11.9 ,1 11.0 11.0 .0 10.3 10.2 .1 2240 11.3 11.0 .3 10.4 10.3 .1 9.7 9.6 .1 2800 10.6 10.3 .3 9.7 9.7 9.2 9.0 .2 3360 10.0 9.6 .4 9.2 9.0 .2 8.6 8.5 .1 3920 9.4 8.9 •i 8.6 8.4 .2 8.1 8.0 .1 4480 8.8 8.3 .f 8.1 7.9 .2 7.6 7.5 .1 5040 8.3 7.8 .5 7.6 7.4 .2 7.2 7.1 .1 5600 7.8 7.3 .5 7.2 7.0 .2 6.8 6.7 .1 12.2 12.3 —.1 11.5 11.6 —.1 12.2 12.2 560 11.5 11.5 10.9 10.9 11.5 11.5 1120 10.8 10.8 10.3 10.3 10.8 10.9 — 1 1680 10.2 10.1 .1 9.7 9.7 10.2 10.3 —.1 2240 9.6 9.5 .1 9.1 9.1 9.6 9.7 —.1 2800 9.0 8.9 .1 8.6 8.6 9.0 9.1 — 1 3360 8.5 8.4 .1 8.1 8.15 —.1 8.5 8.5 3920 8.0 8.0 7.6 7.7 —.1 8.0 8.1 —.1 4480 7.5 7.5 7.1 7.3 .2 7.5 7.7 —.2 5040 7.1 7.0 .1 6.7 6.9 .2 7.1 7.2 —.1 5600 6.7 6.5 .2 6.3 6.4 —.2 6.7 6.8 —.1 303 — 2. Exjierlments with the Brass Sphere and the Short Suspensmt. — Continued. m Computed An Observe' I An 0—0 Computed A^ Observed Computed Observed An 0—0 12.5 12.3 .2 12.8 12.7 .1 12.9 12.8 .1 560 11.8 11.7 .1 12.0 11.95 .1 12.1 12.0 .1 1120 11.1 11.0 .1 11.3 11.3 11.4 11.3 .1 1G80 10.4 10.4 10.7 10.7 10.8 10.7 .1 2240 9.8 9.8 10.0 10.15 —.1 10.1 10.2 —.1 2800 9.2 9.2 9.5 9.5 9.5 9.7 /2 3369 8.7 8.75 —.1 8.9 8.9 9.0 9.1 —.1 3920 8.2 8.3 —.1 8.4 8.4 8.4 8.6 —.2 4480 7.7 7.9 —.2 7.9 8.0 —.1 7.9 8.1 —.2 5040 7.2 7.45 — .2 7.4 7.6 —.2 7.5 7.7 —.2 5600 6.8 7.0 —.2 7.0 7.2 —.2 7.0 7.2 ,2 13.0 12.9 .1 10.9 10.9 13.4 13.2 .2 560 12.2 12.1 .1 10.4 10.3 .1 12.6 12.5 .1 1120 11.5 11.4 .1 9.7 9.7 11.9 11.8 .1 1680 10.8 10.8 9.2 9.2 1J.2 11.2 2240 10.2 10.3 —.1 8.6 8.7 —.1 10.5 10.6 —.1 2800 9.6 9.8 ,2 8.1 8.2 —.1 9.9 9.9 3360 9.0 9.2 —.2 7.6 7.7 —.1 9.3 9.3 3920 8.5 8.8 —.3 7.2 7.2 8.8 8.8 4480 8.0 8.2 —.2 6.8 6.8 8.3 8.3 5040 7.5 7.8 —.3 6.4 6.4 7.8 7.85 —.1 5600 7.1 7.4 —.3 6.0 6.0 7.3 7.45 —.1 13.3 13.3 11.1 11.3 -.2 12.4 12.5 —.1 560 12.5 12.5 10.5 10.5 11.7 11.7 1120 11.8 11.8 9.9 9.8 .1 11.0 10.9 .1 1680 11.1 11.1 9.3 9.3 10.3 10.2 .1 2240 10.4 10.5 —.1 8.8 8.8 9.7 9.6 .1 2800 9.8 9.8 8.3 8.3 9.2 9.0 .2 3360 9.2 9.2 7.8 7.8 8.6 8.5 .1 3920 8.7 8.7 7.3 7.2 .1 8.1 8.0 .1 4480 8.2 8.2 6.9 6.8 .1 7.6 7.6 5040 7.7 7.7 6.5 6.4 .1 7.2 7.2 5600 7.3 7.3 6.1 6.1 6.8 6.8 11.7 11.8 —.1 560 11.1 11.1 1120 10.4 10.4 1680 9.8 9.8 2240 9.3 9.2 .1 2800 8.7 8.7 3360 8.2 8.2 3920 7.7 7.7 4480 7.3 7.3 5040 6.8 6.8 5600 6.4 6.4 304 3. Experh nents with the Ivory Sphere and Long S Hspension. m Computed Obserred C Computed Observed C Computed Observed 0—0 36.5 36.4 .1 38.9 38.9 38.7 38.6 .1 500 21.5 22.0 —.5 22.7 22.7 22.6 22.7 —.1 1000 13.5 13.2 .3 14.2 14.3 —.1 14.1 14.3 —.2 38.9 38.9 37.9 37.8 .1 37.9 37.9 500 22.7 22.9 ,2 22.2 22.6 —.4 22.2 22.4 —.2 1000 14.2 14.5 —.3 13.9 14.3 —.4 13.9 14.0 —.1 39.1 39.2 —.1 37.4 37.5 —.1 38.5 38.5 500 22.7 22.4 .3 21.9 21.7 .2 22.5 22.3 .2 1000 14.2 13.7 .5 13.7 12.9 .8 14.0 14.2 —.2 38.4 38.4 37.0 37.1 —.1 37.3 37.3 500 22.4 22.0 .4 21.7 21.1 .6 21.9 21.8 .1 1000 14.0 14.0 13.6 13.4 .2 13.7 13.9 — 2 37.2 37.3 —.1 36.8 36.8 37.1 36.9 .2 500 21.8 21.7 .1 21.6 21.7 —.1 21.8 22.1 —.3 1000 13.7 13.8 —.1 13.6 13.4 .2 13.7 13.9 — 2 34.7 34.7 500 20.5 20.6 —.1 1000 13.3 13.0 .3 1. Experiments with the Ivory Sphere and Short S uspension. m Computed Observed Computed Observed Computed A,n Observed 12.3 12.3 13.6 13.6 13.9 14.0 —.1 650 9.3 9.3 10.1 10.0 .1 10.3 10.1 .2 1300 7.1 7.2 —.1 7.6 7.8 —.2 7.8 7.8 1950 5.4 5.7 —.3 5.8 5.9 —.1 5.9 5.8 .1 2600 4.2 4.3 —.1 4.4 4.3 .1 4.5 4.3 .2 13.0 13.1 —.1 14.8 14.9 —.1 14.3 14.3 650 9.9 9.9 10.9 10.9 10.6 10.7 —.1 1300 7.5 7.5 8.2 8.0 .2 8.0 8.0 1950 5.7 5.7 6.2 6.0 .2 6.0 6.0 2600 4.5 4.5 4.8 4.5 .3 4.6 4.6 12.9 13.1 .2 14.0 14.0 13.2 13.0 .2 650 9.6 9.6 10.4 10.4 19.8 19.9 —.1 1300 7.2 7.0 .2 7.8 8.0 —.2 7.3 7.4 —.1 1950 5.5 5.4 .1 5.9 6.1 —.2 5.6 5.9 —.3 2600 4.2 4.1 .1 4.5 4.5 4.3 4.4 —.1 13.3 13.1 .2 16.0 16.0 16.8 16.8 650 9.8 10.0 —.2 11.8 11.8 12.4 12.5 —.1 1300 7.4 7.3 .1 8.9 8.8 .1 9.3 9.4 —.1 1950 5.6 5.8 —.2 6.7 6.8 —.1 7.1 7.1 2600 4.3 4.5 —.2 5.2 5.2 5.4 5.5 —.1 305 4. Experiments with the Ivory Sphere and Short Suspension. — Continued. in Computed An Observed C Computed Observed C Computed An Observed An C 16.6 16.6 17.8 18.0 —.2 16.3 16.3 650 12.2 12.1 .1 13.0 13.0 12.0 12.2 —.2 1300 9.2 9.2 9.7 9.5 .2 9.0 9.1 —.1 rjuo 7.0 7.0 7.3 7.1 .2 6.8 7.0 —.2 2600 5.3 5.5 .2 5.6 5.5 .1 5.2 5.2 16.1 10.0 .1 650 11.8 12.0 .2 1300 8.8 9.0 —.2 1950 6.7 6.8 —.1 2600 5.1 5.1 5. Experiments with the Full Cylinder and Long Suspension. m Computed Observed ^4,„ C Computed An Observed r .0 Computed Observed C 39.8 39.8 41.5 41.2 .3 38.3 38.9 —.6 500 35.8 35.9 —.1 37.3 37.3 84.5 34.3 .2 1000 32.4 32.2 .2 33.6 33.8 —.2 31.2 30.8 .4 1500 29.4 29.4 30.5 30.7 —.2 28.4 27.8 .6 2000 26.7 26.6 .1 27.7 27.8 —.1 25.8 25.2 .6 2500 24.3 24.3 25.2 25.3 —.1 23.6 22.9 .7 3000 26.3 22.0 .3 23.0 23.2 —.2 21.6 20.6 1.0 3500 20.4 20.3 .1 21.1 21.3 —.2 19.8 18.7 1.1 4000 18.7 19.0 — 3 19.3 19.5 —.2 18.2 17.1 1.1 39.4 39.6 2 41.0 41.5 —.5 41.7 41.8 —.1 500 35.5 35.3 .2 36.9 36.9 37.5 37.6 —.1 1000 32.1 31.9 .2 33.3 32.9 .4 33.8 33.5 .3 1500 29.1 29.0 .1 30.1 29.9 .2 30.6 30.5 .1 2000 26.5 26.2 .3 27.4 27.1 .3 27.8 27.7 .1 2500 24.1 24.0 .1 25.0 24.6 .4 25.3 25.4 —.1 3000 22.1 22.0 .1 22.8 22.5 .3 23.1 23.2 —.1 3500 20.2 20.0 .2 20.9 20.6 .3 21.2 21.2 4000 18.6 18.3 .3 19.1 19.0 .1 19.4 19.3 .1 39.5 39.2 .3 40.2 40.3 —.1 42.7 42.7 500 35.6 35.6 36.2 3G.1 .1 38.3 38.1 .2 1000 32.1 32.4 —.3 32.7 32.7 34.5 34.6 —.1 1500 29.2 29.4 .2 29.6 30.0 —.4 31.2 31.4 .2 2000 26.5 26.6 —.1 26.9 27.1 —.2 28.4 28.3 .1 2500 24.2 24.3 —.1 24.5 24.7 —.2 25.8 25.9 —.1 3000 22.1 22.3 .2 22.4 22.6 —.2 23.6 23.6 3500 20.3 20.5 —.2 20.6 207 —.1 21.6 21.6 4000 18.6 18.7 —.1 18.9 18.9 19.8 19.9 —.1 39 306 — 5. Experiments with the Fall Ctjlinder and Loncj Suspension. — Continued. m Computed Observed An C—C i Computed An Observed C—0 Computed An Observed An C—0 42.3 42.5 —.2 43.2 43.1 .1 42.0 41.8 .2 500 38.0 37.9 .1 38.8 38.8 37.7 38.4 —.7 1000 34.2 34.0 .2 34.9 35.0 —.1 34.0 34.0 1500 31.0 31.1 —.1 31.6 31.6 30.8 30.9 —.1 2000 28.1 28.1 28.7 28.6 .1 28.0 28.1 —.1 2500 25.6 25.5 .1 26.1 26.1 25.5 25.7 —.2 3000 23.4 23.5 —.1 23.8 23.9 —.1 23.3 23.5 —.2 3500 21.4 21.5 —.1 21.8 21.7 .1 21.3 21.5 —.2 4000 19. G 19.4 .2 20.0 20.0 19.5 19.6 —.1 41.5 41.4 .1 41.4 41.2 .2 41.6 41.4 .2 500 37.3 37.2 .1 37.2 37.3 —.1 37.4 37.4 1000 33.6 33.5 .1 33.6 33.6 33.7 33.8 —.1 1500 30.5 30.5 30.4 30.4 30.5 30.6 —.1 2000 27.7 27.9 —.2 27.6 27.6 27.7 27.9 .2 2500 25.2 25.3 —.1 25.2 25.2 25.3 25.5 —.2 3000 23.0 23.1 —.1 23.0 23.1 —.1 23.1 23.2 —.1 3500 21.1 21.3 —.2 21.0 21.2 —.2 21.1 21.3 .2 4000 19.3 19.4 —.1 19.3 19.1 .2 19.5 19.6 —.1 41.4 41.1 .3 40.5 40.3 .2 39.3 39.4 —.1 500 37.2 37.2 36.4 36.5 —.1 35.4 35.3 .1 1000 33.6 33.6 32.9 33.0 —.1 32.0 32.0 1500 30.4 30.5 —.1 29.8 29.8 29,0 29.0 2000 27.6 27.7 —.1 27.1 27.0 .1 2G.4 26.4 2500 25.2 25.2 24.7 24.6 .1 24.1 24.1 3000 23.0 23.1 1 22.6 22.5 .1 22.0 22.1 —.1 3500 21.0 21.2 2 20.7 20.7 20.2 20.3 —.1 4000 19.3 19.5 2 19.0 19.0 18.5 18.5 38.0 38.3 — 3 39.6 39.5 .1 42.0 41.8 .2 500 34.1 33.5 .6 35.6 35.6 37.7 37.9 — .2 1000 30.9 30.4 .5 32.2 32.1 .1 34.0 34.2 ,2 1500 28.0 27.8 .2 29.2 29.3 — 1 30.8 31.0 .2 2000 25.5 25.3 .2 26.(i 26.7 — 1 28.0 28.2 2 2500 23.2 23.0 .2 24.2 24.3 — 1 25.5 25.6 — 1 3000 21.3 21.3 22.2 22:3 — 1 23.3 23.4 — 1 3500 19.6 19.6 20.3 20.4 — 1 21.3 21.5 — 2 4000 18.0 17.8 .2 18.6 18.7 — 1 19.5 19.6 — 1 42.1 42.0 .1 41.7 41.7 40.6 40.6 500 37.8 37.9 —.1 37.4 37.4 36.5 36.6 —.1 1000 34.1 34.1 33.8 33.9 —.1 33.0 33.0 1500 30.9 30.9 30.6 30.6 29.9 30.0 — .1 2000 28.0 28.1 —.1 27.8 27.9 —.1 27.2 27.2 2500 25.5 25.7 —.2 25.3 25.5 .2 24.8 24.8 3000 23.3 23.3 23.1 23.2 —.1 22.6 22.7 — .1 3500 21.3 21.3 21.2 21.4 —.2 20.7 20.8 — .1 4000 19.5 19.5 19.4 19.6 —.2 19.0 18.9 .1 307 — 6. Experiments rvith the Full Cjjlinder and Short Suspension. m Computed Obseryed An C Computed An Observed An C Computed An Observed An C—0 12.4 12.45 12.1 12.15 13.2 13.15 730 11.7 11.6 .1 11.4 11.35 12.5 12.5 1400 11.0 11.0 10.8 10.65 .1 11.8 11.95 — .1 2100 10.4 10.4 10.2 10.25 —.1 11.2 11.1 2920 9.9 9.8 .1 9.6 9.55 .1 10.5 10.55 — .1 3650 9.3 9.3 9.1 9.1 9.9 10.0 — ,1 4380 8.8 8.85 8.6 8.6 9.4 9.45 — .1 5110 8.3 8.35 8.1 8.15 8.9 8.9 5840 7.9 7.85 7.7 7.65 8.4 8.5 —.1 13.0 13.0 12.1 12.05 12.6 12.55 730 12.3 12.25 11.5 11.7 —.2 1 2.0 12.0 1460 11.6 11.55 10.9 11.05 —.1 11.3 11.3 2190 10.9 11.05 —.1 10.4 10.5 —.1 10.8 10.75 2920 10.3 10.3 9.8 9.95 —.1 10.2 10.1 .1 3650 9.8 9.75 9.5 9.45 .1 9.7 9.65 .1 4380 9.2 9.2 8.9 9.05 —.1 9.2 9.25 5110 8.7 8.8 —.1 8.5 8.6 —.1 8.8 8.8 5840 8.2 8.2 8.0 8.05 8.4 8.35 12.4 12.4 12.4 12.4 13.5 13.5 730 11.8 11.65 .1 11.8 11.75 12.8 12.75 .1 1460 11.2 11.15 .1 11.2 11.2 12.2 12.35 —.1 2190 10.6 10.55 .1 10.6 10.6 11.6 11.65 2920 10.1 10.2 — 1 10.1 10.05 .1 11.0 11.0 3650 9.6 9.6 9.6 9.55 .1 10.5 10.55 4380 9.2 9.15 9.2 9.15 .1 10.0 10.0 5110 8.7 8.7 8.7 8.65 .1 9.5 9.55 5840 8.3 8.35 8.4 8.3 .1 9.1 9.1 13.0 12.95 13.6 13.6 13.9 14.0 —.1 730 12.4 12.4 12.9 13.0 —.1 13.2 13.2 1460 11.7 11.85 — 1 12.3 12.05 .2 12.5 12.65 —.1 2190 11.2 11.3 — 1 11.6 11.5 .1 11.9 11.95 2920 10.6 10.8 2 11.1 11.0 .1 11.3 11.4 —.1 3650 10.1 10.1 10.5 10.55 10.7 10.55 .2 4380 9.7 9.7 10.0 10.0 10.2 10.25 5110 9.2 9.2 9.5 9.55 9.7 9.8 —.1 5840 8.8 8.9 —.1 9.0 9.0 9.3 9.2 .1 — 308 7. Experiments with the Emptij Cijlinder and Long Suspension m Computed Observed C Computed An Observed C Computed Observed C 37.5 37.7 —.2 38.2 37.8 .4 37.6 37.6 500 28.1 27.7 .4 28.6 28.8 .2 28.2 28.3 .1 1000 21.4 21.0 .4 21.8 22.0 —.2 21.5 21.4 1500 16.5 16.6 —.1 16.8 16.7 .1 16.6 16.7 — .1 2000 12.9 13.1 — .2 13.1 13.0 .1 12.9 13.0 —.1 38.0 37.9 .1 39.2 38.8 .4 38.9 38.8 500 28.4 28.6 —.2 29.3 29.4 —.1 29.1 29.2 — ,1 1000 21.7 21.6 .1 22.3 22.4 —.1 22.1 22.3 — .2 1500 1G.7 16.8 —.1 17.2 17.2 17.1 17.2 • •'■ 2000 13.0 13.0 13.4 13.4 13.3 13.2 40.3 40.4 —.1 40.8 40.8 38.0 38.0 500 30.0 29.9 .1 30.4 30.4 28.4 28.4 1000 22.8 22.5 .3 23.1 23.1 21.7 21.7 1500 17.6 17.8 —.2 17.7 17.9 —.2 16.7 16.8 —.1 2000 13.7 13.9 —.2 13.8 13.9 —.1 13.0 13.0 37.9 37.9 40.2 40.2 39.9 40.0 —.1 500 28.4 28.4 30.0 30.0 29.7 29.6 .1 1000 21.6 21.5 .1 22.8 22.9 —.1 22.6 22.5 .1 1500 16.7 16.8 —.1 17.5 17.6 —.1 17.4 17.3 .1 2000 13.0 13.0 13.6 13.9 —.3 13.6 13.6 39.8 40.0 2 39.2 39.1 .1 40.7 40.5 .2 500 29.7 29.7 29.3 29.6 —.3 30.3 30.4 —.1 1000 22.6 22.4 .2 22.3 22.3 23.0 23.1 —.1 2500 17.4 16.7 .7 17.2 17.2 17.7 17.6 .1 2000 13.5 13.2 .3 13.4 13.4 13.8 13.7 .1 40.4 40.4 40.1 40.2 —.1. 40.4 40.4 500 30.1 30.3 —.2 29.9 29.9 30.1 30.2 —.1 1000 22.9 22.7 .2 22.7 22.7 22.9 22.7 .2 1500 17.6 17.5 .1 . 17.5 17.5 17.6 17.6 •0 2000 13.7 13.7 13.6 13.6 13.7 13.6 .1 38.9 38.9 38.6 38.8 2 38.7 38.7 500 29.1 29.1 28.8 28.6 .2 28.9 28.8 .1 1000 22.1 21.9 .2 22.0 21.9 .1 22.0 22.2 — .2 1500 17.0 17.0 16.9 16.9 17.0 17.1 —.1 2000 13.3 13.4 — 1 13.2 13.2 13.2 13.4 .2 39.5 39.5 38.1 38.0 .1 38.5 38.5 500 29.5 29.8 —.3 28.5 28.6 —.1 28.8 29.0 — 2 1000 22.4 22.4 21.7 21.8 —.1 21.9 21.9 1500 17.3 17.2 .1 16.8 16.6 .2 16.9 16.8 .1 2000 13.4 13.4 13.0 12.9 .1 13.2 13.0 .2 — 309 — 7. Experiments with the Empty Cylinder and Long Suspension. — Continued. m Computed Observed C Computed An Observed An C—0 Computed Observed An C—0 39.2 38.8 A 38.5 38.4 .1 39.0 39.0 500 29.3 29.5 —.2 28.8 29.0 —.2 29.1 29.2 —.1 1000 22.3 22.4 —.1 21.9 21.9 22.2 22.2 1500 17.2 17.2 16.9 16.9 17.1 17.2 —.1 2000 13.4 13.6 .2 13.2 18.2 13.3 13.4 —.1 39.9 39.9 38.4 38.4 38.9 38.9 500 29.8 29.8 28.7 28.8 — 1 29.1 29.2 —.1 1000 22.6 22.8 — .2 21.9 22.0 1 22.1 22.1 1500 17.4 17.6 — .2 16.9 17.0 _.l 17.1 17.1 2000 13.6 13.6 13.1 13.1 13.3 13.2 .1 38.6 38.6 39.1 39.1 36.8 36.8 500 28.9 28.9 29.2 29.3 1 27.6 27.6 1000 22.0 21.9 .1 22.2 22.1 .1 21.1 21.1 1500 16.9 16.8 .1 17.1 17.0 .1 16.3 16.3 2000 13.2 13.1 .1 13.3 13.1 .2 12.6 12.6 36.3 36.3 38.3 38.3 39.4 39.3 .1 500 27.3 27.4 1 28.6 28.7 1 29.4 29.5 —.1 1000 20.8 21.0 .2 21.8 21.8 22.4 22.4 1500 16.1 16.1 16.8 16.8 17.2 17.3 —.1 2000 12.5 12.5 13.1 13.1 13.4 13.5 —.1 40.1 40.1 39.4 39.3 .1 38.8 38.8 500 29.9 29.9 29.4 29.5 — 1 29.0 29.4 —.4 1000 22.7 22.6 .1 22.4 22.4 22.1 22.0 .1 1500 17.5 17.4 .1 17.2 17.3 — 1 17.0 17.0 2000 13.6 13.5 .1 13.4 13.5 — 1 13.2 13.3 —.1 40.0 40.0 38.7 38.7 38.8 38.9 —.1 500 29.8 30.0 /2 28.9 28.9 29.0 28.9 .1 1000 22.7 22.5 .2 22.0 21.9 .1 22.1 22.0 .1 1500 17.4 17.3 .1 17.0 16.9 .1 17.0 17.1 —.1 2000 13.6 13.6 13.2 13.2 13.2 13.3 —.1 39.1 39.1 38.5 38.5 38.7 38.7 500 29.2 29.1 .1 28.8 28.8 28.9 29.0 — 1 1000 22.2 22.1 .1 21.9 21.8 .1 22.0 22.0 1500 17.1 17.1 16.9 16.9 17.0 16.9 .1 2000 13.3 13.3 13.2 13.1 .1 13.2 13.2 37.7 37.7 38.8 38.8 38.1 38.2 —.1 500 28.2 28.0 .2 29.0 29.0 28.5 28.5 1000 21.5 21.5 22.1 22.0 .1 21.7 21.5 .2 1500 16.6 16.7 —.1 17.0 16.9 .1 16.8 16.6 2 2000 12.9 12.9 13.2 13.1 .1 13.0 12.9 .1 — 310 8. Experiment^: ivith the Empti/ Cijlinder and the Shm-t' Suspension. m Computed An Observed C—0 Computed Observed ^ /-^ Computed Observed ' Aa C—0 11.4 11.4 12.2 12.3 — 1 13.3 13.3 800 9.0 9.4 .1 10.1 10.15 11.0 10.95 IGOO 7.9 7.7 2 8.5 8.45 9.1 9.05 2400 6.6 6.5 1 7.1 7.05 7.6 7.55 3200 5.5 0.5 .0 6.0 5.9 .1 6.3 6.3 4000 4.7 4.6 .1 5.0 4.9 .1 5.3 5.3 4800 4.0 3.9 .1 4.2 4.1 .1 4.4 4.5 —.1 13.3 13.3 13.4 13.4 12.1 12.1 800 11.0 11.1 —.1 11.3 11.3 10.2 10.25 IGOO 9.1 9.15 9.5 9.4 .1 8.7 8.65 2400 7.6 7.8 —.2 8.0 8.05 7.3 7.4 —.1 3200 6.3 6.2 .1 6.8 6.7 .1 6.2 6.25 4000 5.3 5.4 —.1 5.8 5.7 .1 5.3 5.3 4800 4.4 4.45 4.9 4.9 4.5 4.5 12.3 12.2 .1 13.0 13.0 13.2 13.15 800 10.5 10.5 11.0 11.0 11.2 11.15 1000 8.9 9.0 —.1 9.4 9.3 .1 9.5 9.6 — 1 2400 7.6 7.85 2 8.0 8.0 8.1 8.0 .1 3200 6.5 6.7 ,2 6.9 6.9 6.9 6.85 4000 5.6 5.75 ,2 5.9 5.9 5.9 5.95 . 4800 4.8 4.95 —.1 5.0 5.1 —.1 5.1 5.1 13.0 12.95 14.1 14.25 —.1 12.9 12.9 800 11.0 11.05 11.9 11.9 10.9 10.95 IGOO 9.4 9.4 10.1 10.1 9.3 9.15 .1 2400 8.0 8.05 —.1 8.6 8.65 —.1 7.9 7.85 3200 6.8 6.9 —.1 7.3 7.35 —.1 6.7 6.85 —.1 4000 5.8 5.85 6.2 5.95 .3 5.7 5.85 —.1 4800 5.0 5.0 5.3 5.25 .1 4.9 4.9 In the computation of these values, there has been no regard to the resistance arising from the wires of suspension. The dif- ference between the vahies of ^ may be attributed to the uncer- tainty of the observations, and those of H^ niay, perhaps, be ac- counted for, in the same way. The vahie of H^ is nearly ten times as great as that Avliich is given by the observations of Borda upon the resistance of the atmosphere. It must, therefore, be doubtful, whether the observed diminution of the arcs of vibration — 311 — of the pendulum is, wholly or principally, clue to the medium in which it vibrates, or to some more latent cause. This doubt is much increased by the discussion of the observations of Baily. 534. In Baily's experiments, various pendulums, which were mostly spheres and cylinders, were vibrated in the receiver of an air-jDump, with the air either at its ordinary pressure, or at the small density of about one thirtieth of an atmosphere. For the full and exact description of the pendulums the original memoir must be consulted, but the following brief description is sufficient for the present purpose. Numbers 1, 2, 3, and 4 are spheres of platina, lead, brass, and ivory, all of the same diameter, which is somewhat less than 1 J inches, and of which the weights with their vibrating appendages are, respectively 9050, 4648, 3217, and 776 1 grains. Nos. 5, 6, and 7 are spheres of lead, brass, and ivory, all of the same diameter, which is 2.06 inches, and of which the weights are respectively, 13019, 9302, and 2066^ grains. Nos. 8 and 9 are the same spheres of lead and ivory with those of Nos. 5 and 7, but suspended from a wire passing over a small cylinder instead of from a knife edge. In Nos. 10, 11, 12, and 13 the vibrating mass was a brass cylinder, of which the diameter of the base is 2.06 inches, the altitude 2.06 inches, and the weight 14190 grains ; in Nos. 10 and 13 the axis of the cylinder coincides with that of the pendulum rod, but the rod of No. 13, which was also adopted in Nos. 11 and 12, w^as a thick brass wire 0.185 inch in diameter, 37^ inches long, and weighing 2050 grains 5 in Nos. 11 and 12 the axis of the cylinder was horizontal, in No. 11 it was perpendicular to the plane of vibration, and in No. 12 it was in the plane of vibration. No. 14 is a cylinder of lead, of which the diameter of the base is 2.06 inches, the altitude 4 inches, the weight 34500 grains, and the axis coincident w4th the rod of the pendulum. In Nos. 15, 16, 17, 18, and 19 the vibrating mass was a hollow cyl- — 312 — inder of the same position and external dimensions with No. 14 ; in No. 15 both ends were open ; in No. 16 the top was open and the bottom closed; in No. 17 the top was closed and the bottom open; in No. 18 both ends were closed ; in No. 19 an inner sliding tube was removed so as to reduce the weight; and the weights, with the inclosed air, were, respectively, 8497, 8922, 8622, 9048, and 7250 grains. No. 20 is a lens of lead 2.06 inches in diameter, an inch thick in the middle, with a flat circumference of about a quarter of an inch wide, and a weight of 6505 grains. No. 21 is a solid copper cylindrical rod of 0.41 inch in diameter, 58.8 inches long, and weighing 16810 grains. In Nos. 25, 26, 27, 28, 29, 30, 31, 32, 33, and 34, the vibrating masses were convertible pendulums, formed of plane bars, and they are vibrated successively with each of their points of suspension, which were knife edges; in Nos. 25 and 26 the bar was brass, two inches wide, three eighths of an inch thick, 62.2 inches long, and weighing 121406 grains; in Nos. 27 and 28 it was copper of the same width with the brass bar, half an inch thick, 62.5 inches long, and weighed 155750 grains; in Nos. 29 and 30, it was iron of the same width and thickness with the copper bar, 62.1 inches long, and weighed 140547 grains; in Nos. 31, 32, 33, and 34 it was a doubly convertible brass bar, three quarters of an inch thick, 62 inches long, and weighed 231437 grains. In Nos. 35, 36, 37, and 38, a doubly convertible pendulum, made of a brass cylin- drical tube of Ij inches in diameter, 56 inches long, and weighing 81047 grains was vibrated upon a knife edge with all four of its planes of suspension. No. 39 is a mercurial pendulum. Nos. 40 and 41 are clock pendulums in which the vibrating mass was a leaden cylinder 1.8 inches in diameter, 13.5 inches long, and weighing 93844 grains ; in No. 40 it was suspended from a spring, by a cylin- drical rod of deal of three eighths of an inch in diameter, and in No. 41 by a flat rod of deal one inch wide, 0.14 inch thick in the — 313 — middle of its width and bevelled on each side to a thin edge, which was opposed to the direction of its motion. In the discussion of Baily's experiments, the value of II2 is neglected, because it is of small influence, and the arcs of vibration, being usually given only for the beginning and end of the experi- ment, are just sufficient to determine one of the quantities H^ and H2 ', and the values of H^ are not reduced to the same density of air. The ratio of the value of Hi for the ordinary state of the air to its value in the exhausted receiver, varies from 1.9 to 4.2, in- stead of being about 30, which it should be if it were proportional to the density of the air ; the value of this ratio in the following table is expressed by J. The total resistance to the motion of the pendulum, supposed to be proportional to the velocity is, for the imit of velocity, expressed by H[' in the table ; and this same re- sistance, reduced to the unit of Aveight, is expressed by Hi. The observation of the arcs of vibration in Baily's experiments is limited to the initial and final arcs, and the direct comparison of the computed and observed arcs is, consequently, quite unnecessary, and cannot contribute to verify the accuracy of the hypothesis upon which the computation is based. The only two cases in which an intermediate arc was observed with Nos. 6 and 14 seem to sustain the hypothesis ; for they differ from it slightly, but in oj^posite direc- tions. The diversity of the values of H^ indicates that the resisting force of the motion to the pendulum demands a new experimental investigation, conducted with a direct object to its determination ; and that, until such an investigation has been made, the length of the seconds pendulum must be regarded as liable to an unknown error. 40 314 Values of Hi in Bailij's Experiments upon the Vibrations of Pendulums. No. of PenJulums. Barometer. H, Hi m' J 1 0.7089 .0673 .000077 .000132 2.68 3 0.7646 .0702 .000080 .000384 2.62 2 0.7523 .0662 .000075 .000250 2.55 4 0.7660 .0561 .000063 .001272 2.71 6 0.7638 .0570 .000123 .000204 2.74 7 0.7630 .0538 .000116 .000864 2.62 5 0.7644 .0627 .000128 .000161 3.18 9 0.7(;82 .0589 .000127 .000945 2.86 8 0.7677 .1021 .000219 .000261 2.92 10 0.7652 .0651 .000179 .000194 3.42 11 0.7637 .0558 .000270 .000256 2.62 12 0.7623 .0603 .000290 .000277 3.33 13 0.7552 .0571 .000235 .000262 2.98 18 0.7491 .0535 .000285 .000484 3.27 15 0.7554 .0658 .000350 .000635 4.10 16 0.7495 .0595 .000292 .000505 2.95 17 0.7584 .0558 .000297 .000531 3.39 14 0.7747 .0592 .000315 .000140 4.22 19 0.7620 .0510 .000272 .000578 3.33 20 0.7620 .0656 .000065 .000156 2.09 21 0.7575 .0661 .000742 .000682 2.72 25 0.7522 .0789 .005606 .000333 3.32 2G 0.7465 .0756 .004782 .000319 3.74 31 0.7522 .1555 .003666 .000245 3.32 32 0.7520 .1581 .003673 .000245 3.55 34 0.7529 .1661 .003772 .000251 3.72 33 0.7535 .1417 .003480 .000232 3.13 35 0.7595 .0739 .003091 .000589 3.48 36 0.7627 .0660 .002763 .000526 3.31 37 0.7577 .0701 .002931 .000558 3.39 38 0.7564 .0659 .002760 .000526 2.97 39 0.7622 .001396 .000209 1.87 41 0.7573 .0664 .001260 .000207 2.52 40 0.7589 .0769 .001299 .000213 2.39 — 315 — Values of III '" Bailij's Exper'nnonts upon the Vibrations of Pendulums. — Continued. Xo. of Pendulums. Barometer. H, m Hi' J 1 0.0288 .0251 .000028 .000049 2.68 3 0.0294 .0267 .000031 .000146 2.62 2 0.02 Go .0259 .000030 .000098 '2Jro 4 0.0347 .0284 .000024 .000470 2.71 6 0.0268 .0285 .000044 .000074 2.74 7 0.0270 .0282 .000044 .000330 2.62 5 0.0290 .0275 .000042 .000050 3.18 9 0.03 GO .028.2 .000044 .000331 2.86 8 0.0299 .0348 .000075 .000089 2.92 10 0.0239 .0190 .000052 .000057 3.42 11 0.0478 .0213 .000103 .000098 2.62 12 0.0348 .0182 .000088 .000083 3.33 13 0.0370 .0192 .000092 .000O.S9 2.98 18 0.0300 .0164 .000087 .000148 3.27 15 0.0271 .0164 .000097 .000148 4.10 - 16 0.02 66 .0186 .000099 .000171 2.95 17 0.03 G2 .0165 .000088 .000157 3.39 14 0.0298 .0139 .000074 .000033 4.22 19 0.0305 .0154 .000083 .000174 3.33 20 0.0305 .0313 .000031 .000074 2.09 21 0.0288 .0244 .000274 .000251 2.72 25 0.0313 .0238 .001505 .000101 3.32 26 0.0325 .0202 .001277 .000086 3.74 . 31 0.0414 .0469 .001105 .000074 3.32 32 0.0391 .0439 .001034 .000069 3.55 34 0.0410 .0431 .001014 .000067 3.72 33 0.0463 .0472 .001111 .000074 3.13 35 0.0384 .0213 .000888 .000170 3.48 36 0.0367 .0200 .000834 .000160 3.31 37 0.0422 .0206 .000859 .000166 3.39 38 0.0412 .0222 .000930 .000178 2.97 39 0.0477 .000747 .000112 1.87 41 0.0457 .0263 .000498 .000083 2.52 40 0.0434 .0320 .000543 .000089 2.39 — 31G THE TAUTOCHRONE. 535. The consideration of the pendulum leads, directly, to the investigation of that curve, upon which the duration of the vibration is independent of the length of the arc of oscillation. Such a curve is called a iaidochrone, and is readily determined when the body is only subject to the action of fixed forces. 536. If the force which acts in the direction of the motion of the body is denoted by S, the equation of its motion is In the case in which /S' is a function of s, let Sq denote the point, at which the velocity vanishes, or the extremity of the arc of vibration. Hence v' = 2Jy=.2{n-ll,)- and if the origin of coordinates is at the point of maximum velocity, the time of vibration is determined by the equation If h — - Sn if i2 is a function of s expressed by X2,, and if s is written instead of ^0, the value of T becomes In order that the special value of the arc may disappear from — 317 — this integral, it is obvious that S2, has the form which reduces the value of T to Jhs/Bs/il — fr) \2B The tangential force along the curve is, therefore, 537. If ^ denotes the actual force, which acts upon the body in the direction of/, the preceding equation gives for an equation of the taiitochrone Fco8{^ — 2Bs=FI)J, or A—Bs^=zJf. 538. In the case in ivhich the hody is restricted to move ttpon a curve ivhich rotates uniformly about a fixed axis, the equations and notation of §468 combined with the previous section, give for the equation of the tautochrone A — Bs''=la''u'', which may assume the form S' , u -4-- — 1 in which a and b are constants. 539. When the revolving curve is a plane curve, and situated in the same plane with the axis of revolution, the notation b =:■ a cot i, s :=:z a sin ^=^ a sin (p sin /, — 318 — and that of elliptic functions give u^h cos ^ , z = -. — : ^i (f — b cos i 73 i (p and if t is the inclination of the curve to the axis of rotation, its value is sin T = — cot i tan 6 . The maxiniuni of ii is b, but its least value, corresponding to or ^ = Ijl i, is 11 = b cos i ; and the corresponding value of s is s=:ljlasin/. The curve consists of several branches, which form cusps by their mutual contact at their extremities, and it resembles the cycloid in its general character. 540. In the case of a heavy body moving iqwn a plane vertical curve, let V denote the angle which the radius of curvature q makes with its horizontal projection, and the equation (SIT^) gives ^z= — ^COSl^, which is the equation of the cycloid referred to its radius of curva- ture and angle of direction, so that the cycloid is the tautochrone of a free heavy body in a vacuum. The same curve, drawn upon the de- — 319 — veloped surface, is the taidochrone of a heavy lodfj, moving upon a vertical cylinder. 541. Every curve may be regarded as being upon the surface of its vertical cylinder of projection ; and, therefore, the taidochrone of a heavy hody moving in a vacuum upon any surface luhatever, is the intersection of the surface with such a vertical cylinder, that the intersection is a cycloid upon the developed vertical cylinder. The determination of the tautochrone upon any surface is thus reduced to a problem of pure geometry. If the axis of z is the upward vertical, and if z^ is the height of the lowest point of the curve above the origin, the equation (SlTie) becomes, in the present case, 542. If a heavy hody is restricted to move upon a cylinder of which the axis is horizontal, and of which the equation of the base is ()j = na cos Vj sin""-^ v-^ , in which v-^ is the angle, which the radius of curvature, denoted by ()i, makes with the upward vertical ; and when the cylinder is devel- oped into a vertical plane, if y is the height of the moving body above the horizontal line, which corresponds to the lowest side of the undeveloped cylinder, the value of y is y ■=.a sin " i\ . The force of gravity, resolved in a direction tangential to the cylinder, is ^sinr'i==yy/^; so that the present problem, corresponds to that of a hody moving in a ver- — 320 — tical plane, and subject to a force ivJiich is fixed in direction, and propor- tional to some poiver of the height above a given level. The equation (319i3) gives for the equation of the tautochrone 543. If V denotes the angle which the radius of curvature [{)) of the tautochrone makes with the upward vertical in the developed cylinder, the equation (317i4) gives 2B sm V sin v, = — s. 9 which, substituted in (32O5), reduces the equation of the tautochrone to g ' n-\-l \gsva. v/ 544. When ^0 vanishes in the problem of the preceding sec- tion, the equation of the tautochrone becomes , . 1+1 I naq /y\V±. n + l n+l , . -^ = — ~ sni"-^ V or () = — —- sni"-^ V cos v ^ n — 1 in which (^^r-'='^m so that the tautochrone on the developed cylinder of § 542 is of the same trigonometric class of curves uith the base of the cylinder, U'hen it passes through the lowest side of the undeveloped cylinder. This case is impos- sible, when n is included between positive and negative unity ; for when n is negative and, independently of its sign, less than unity, s becomes infinite when y vanishes, but when n is positive and less — 321 — than unity, the derivative of (32O19), which is D^S = COSeC V = ^ y/ -jj [-) -An , gives the impossible result that cosec v vauislies with ?/. 545. The differential equation of the tautochrone, in the case of § 542, referred to rectangular coordinates upon the developed cylinder, is readily obtained from the equations of § 542, which give in which 7,2 " + ^ and the axis of x is horizontah In the case of § 544, in which ^^ vanishes, this equation becomes /?,.^+l=/r(f)--\ 54G. In the case in which n is unit}', that is, in which the base of the cjjlinder is a ctjcloid, the equation of the tautochrone on the developed cylinder, becomes AYhen z^ vanishes, this curve is reduced to a straight line, but in all other cases, its form, if it is infinitely extended in the plane of the developed cylinder, resembles the hyperbola. By the adop- tion of the notation . ^ . 2aB snr I =^ , it y=^{2az,)Bec(p, 41 322 and that of elliptic functions, its equation may be expressed in the forms ^ = y/^tan9), :r = J^-jj (cos & tan 9) + ^z 9 — ^i 9) • 547. If ci heavy hody is restricted to move upon a surface of revo- lution about a vertical axis, of which the equation of the meridian curve is that of (319i-). If ^ is the distance of the body on the meridian curve from the lowest point of the surface, the value of y is given by the equation (olOgs), and the force of gravity, resolved in a direction tangential to the meridian curve is expressed by (SlOgg), so that the present problem resembles that of a body moving in a plane, and subject to a force, which is directed towards a fixed point in the plane, and is proportional to some power of the distance from that point. The equation (olT^) of the taidochrone, gives f, ,,m + l ,, jn + 1 p ,.2 9 y — !h m -j- 1 a'" ' in which m is the reciprocal ofn, and y^ the value of y at the lowest point of the tautochrone. 548, When m vanishes, the surface of revolution is a rir/ cone, and the equation (322j9) becomes Bs''==:y{y—y,). By means of the notation sec p =: 1 H -; the angle ((p) which ?/ makes with ?/q in the developed cone is given by the formula tan[(0 + jy)tani|<(] = ^; SO that the polar equation of this tautochrone upon the developed cone is expressed hy the comhinaiion of (32228) and (323o). 549. When ^^ vanishes, /i also vanishes, and the equation (3233) becomes ^ + ^y + cot^=o. 550. When m is unity, the surface of revolution is cj'cloidal and the equation (322j9), becomes which becomes the meridian curve itself, when y^ vanishes. •551. In the case given in (322^4), of a body moving in a plane and subject to a force, ivhich is directed totvards a fixed point in the plane, and is proportional to some potver (m) of the distance from that jjoini, the equation of the tautochrone may he given in the form in which the attracting point is the origin of polar coordinates. The polar differential equation is 552. If the attraction or repulsion of the point had hecn any function zvhatever of the distance from the origin, the equation of the tautochrone tvoidd have assumed the form s^^Fr — Fr 0? — 324 — in Avliicli F denotes the function of which the derivative expresses the given law of attraction. This equation may therefore assume the form in which S-^ is a function of S. If then v is the angle which the radius of curvature makes with the axis of x, the derivatives of this equation are 2 X sin V — 2y cos v=z S[, (2^ cos v-\-2i/ sin v) D,v = S['— 2 • whence 2xD,v = S[ sin v B.v -\- {jS[' — 2) cos v, 2^ B.v = — >S[co^v B.v + {jS[' — 2)smv, iS^Dy-=StDy-^{S" — 2f, Avhich is the equation of the taidochrone expressed in terms of the radius of curvature and the arc. 553. The polar differential equation of the tauiochrone in the case of the preceding section is r'D.if + l Fr—Fu' which is the same equation with that wdiich is given by Puiseux. 554. The derivative of (324ig) relatively to v is so that the _ elimination of 5 between (324iq) and (32427) gives the differential equation of this tauiochrone in terms of the radius of curvature and the angle of its direction. — o-::o — 555. In the case of § 552, when /S ] =^ as -\- b, the value of S^ is The eqiiation (32427) becomes therefore and the cqiiaUon of the taidochrone is ^) = ^ av, 21'hicJi is that of the involute of the circle. This case corresponds to that in which the law of the central force is of the form 556. In the case of § 552, when >Si = « (5 -f- by, the value of S^ is S,= i/^m^{s + ^)\ in which n^ a I— a SO that cc must be positive and less than unity. The equation of the taidochrone is, then,, which is that of the logarithmic spiral. This case corresponds to that in wdiicli the law of the central attraction is of the form — 326 — that is, in ivlikli the force is proportional to the distance of the Ijody from the circumference of the circle described from the origin as the centre ivith a radius erpial to that of the initial position of the hody. Tliis case is discussed by Puiseux. 557. In the case of § 552, ivhen the force is proportional to the distance from the origin. The equation (3233i) assumes the form .2 _ r'—rl a ' which, with the value of m in (32522), reduces S^ and S^ to The equation of the tautochrone is, therefore, of which the integral is Q = , " Cos (m v) ^ 1 — a ^ ' in which the arbitrary constant is determined so that v may vanish with s. The second derivative of this equation gives, for the radius of curvature of the second evolute of the tautochrone so that the second evolute is similar to the tautochrone itself In the case in ivhich m is real, which corresj^onds to that in 327 which a is positive and less than unity, this curve runs off to infinity in each direction, with a constantly increasing radius of curvature. In the case in ivhich m is imaginavfj, the substitution of 2 2 — • n = ntr , reduces the equation of the tautochrone to the form ()= —-^cos (nv). zvJiich is the equation of an epicycloid. The epicycloid is formed hy the external rotation of one circle upon another, ivhen n is less than unity, in ivhich case a is negative and the force is repulsive ; hut the epicycloid is formed hy internal rotation, when n is greater than unity, u'hich corresponds to the case when a is positive and greater than unity. In either of these cases, the initial velocity must not be more than sufficient to carry the body to either of the cusps. In the case in ivhich a is infinite, the tautochrone is reduced to a straight line. The example of this section is discussed by Puiseux. 558. The examj^le of the preceding section embraces the case of any force, which is a function of a distance from the origin, in the immediate vicinity of the point of greatest velocity. The form of the tautochrone, near the point of greatest velocity, in the example of § 552, is typified, therefore, hy the epicycloid, or hy the curve of erpia- tion (32621). 559. The hivestig-ation of the tautochrone in a resisting; medium is postponed to the general case of the chronic curves. — 328 THE BRACHTSTOCHRONE. 5G0. The curve upon wliicli a body moves in the least pos- sible time from one given point to another, is called the hrachjs- tochrone. 561. The investigation of the general case of a brachysto- chrone which is confined to any surface or limited by any condition, may be conducted by means of rectangular coordinates. The time of transit from the first to the last of the given points may be ex- pressed by the equation which is to be a minimum. This condition gives, for each of the other axes, the equation A(^)-Ai>,.(^) = o. 562. When the body is only subject to the action of fixed forces, V does not involve either t/' or /, and the preceding equation becomes D,. or by (316i;), i),i2 + t.^A(^')=0. 563. If the plane of x i/ is assumed, at each instant, to be that in which the body moves, and if the axis of u is taken normal — 82y — to the path of the body, the preceding equation becomes, if o ex- presses the radius of curvature of the path so that ilie ccninfvgal force of iJie hodij h equal to the normal jivcssnre, and the whole 2yi^^ssi[re upon the hrachf/stochroue is double the centrifugal force. This proposition was discovered by Euler. 06 4. When the normal pressure vanishes, the radius of curvature is infinite, which corresponds in general to a point of contrar// flexure. - When there is no force acting upon the hodg throughout its patli, the hrachgstochrone is reduced to a straight line. 565. Any conditions to which the path must be subject, whether elementary such as that it is confined to a given sur- face, or integral such as that its whole length is given, must be combined with the general condition of brachystochrouity hy the usual methods of the calculus of variations, 566. If the only force u'hich acts upon the hodg is directed to a given point, and if the path is sul)ject to no conditions, \Qt the plane of .z-.e- be assumed to be that which passes through the centre of action and the initial element of the path. In this case the equation (02827) gives cos^ = 0, ^=^:t, or the hrachgstochrone is contained in a plane v:hich passes through the centre of action. 567. The preceding case includes that in which the centre of action is removed to an infinite distance, so that, in the case of 'parallel forces, the free hrachgstochrone is contained in a plane, which is parallel to the direction of the forces. 568. When the hodg is acted upon hy no forces, or only hy those ivhich are normcd to its path and do not tend to change its velocity, the 42 — 330 — equation (32813) shoivs that the hr achy dochr one is the shortest line which can lie drawn under the given conditions. 569. When the force is directed towards a fixed centre, the equation (320,,), combined with (31Gi8) gives, if the centre is adopt- ed as the origin D,^ 2 If p is the perpendicular let fall from the origin upon the tangent to the curve, this equation becomes Si — Sio ^"^^i" of which the intcixral is wliich is the equafion of the hrachystochrone referred to the radius vector and the loerpendicular front the origin ujwn the tangent as the coordinates. This form is given by Euler. 570. When the force in the preceding case, is proportional to the distance fro)n the origin so that 11 has the form n^ar'', the equation (33O14) becomes of which the derivative jj;ives P If V is the angle which {) makes witli the fixed axis, the de- ooi rivative of this last equation gives, by means of the preceding equation ''A ^v V = '' cos :=\ [_'(}r,{\ — af,) o- + rfj , which becomes if 2 1 — a pi a PI The integral of this equation is =: — —, Sin (ni v) ^ m u j)i ^ ' SO that its second cvoluic is similar to the hrach/jstochronc itself. When m is real, which corresponds to the case of a repulsive force, and ap\ less than unity, this hrach>/stoehrone is a spiral which has a ciisjj at the point at n'hivh v vanishes. When m is imaginary, the substitution of (32T5) reduces (331n) to the real form ■7. sin {n v) napi so that in this case, the hrachystochrone is an epicycloid jrhich is formed hj internal rotation when the force is attractive, and hjj external rotation when the force is repulsive. This case is given by Euler. 571. ^yhcn the forces are p>arcdlel, ihQ equation (0293) gives, if the axis of * is supposed to be in the direction of the forces ^'^ =4,=2cot=A:, ^ ^^, QSm of which the integral is r> n -^0 in which a is an arbitrary constant, and this is the equation of the hracht/stochrone referred to the coordincdes, ivhich are z and the inclina- tion of the curve to the axis of z ; and the equation, referred to q and I as coordinates, is ohtaincd hy eliminating z hetween (03I27) and (33I31). 572. In the case of a constant force, the preceding equation assumes the forms (J (z — z^) = a sin 2« . ^ so that, in this case, the hrachystochrone is a cycloid. 573. ^Yhen the imrallel forces arc iwoportional to the distance from a given line, which may be adopted for the axis of x, the vaUie of il has the form wdience the equation of the hrachystochrone is a sin s When the force is repulsive, or luhen it is attractive, hut this curve consists of hranches, ivhich are united hy cusps, and resemhle the cycloid in general form ; hut tvhen the force is attractive, and this curve consists of hranches which are still united hy external cusps ; hut the middle point of each hranch is upon the axis of x, and is a point of inflexion,) and the interval hctiveen tivo successive points of inflexion, cx- jjressed hij elliptic integrals, is v/(-i)[^.(i^0-9^.(J^)], in Avliich sin 1:=. z,^\J , ^ a In the case of the attractive force, and I a the equation of the Irachgstochronc becomes () — .ejan^, tvhich consists of two inflnite branches joined hj an external cusp, and the axis of X is an asymptote to each of the branches. 574. When the body is subjected to move upon a given sur- face, the force by which it is retained upon the surface is joerpen- dicular to its path, and must be united with the second member of equation (8293). Hence it follows that the centrifugal force of the body in the direction of the tangent plane to the surface, upon ivhich it is conflned, is equal to the normal force ivhich acts in this plane normal to the brachjstochrone. At the beginning of the motion when the velocity is zero, there is no centrifugal force, so that the initial direction of the br achy stochr one upon the surface coincides vAih that of the tangential force. 575. If the first and last points of the brachystochrone are so situated upon the given surface, that a line can be drawn through them, which coincides throughout with the direction of the tangential force to the surface, this line is the brachysto- chrone. OO-i Hence, the hrach/jstochrone upon the surface of revolution is the meridian line, ivhen both its extremities are upon the same meridian line, and the force is directed to a point upon the axis of revolution, or is parallel to this axis. 576. In the general case of a surface of revolution and a force which is directed to a point upon the axis of revo- lution, let o denote the arc of the meridian curve measured from the pole, u the perpendicular from the surface upon the axis, ^)^ the radius of curvature of the projection of the brachjstochrone upon the tangent plane to the surface, and the proposition (SoSgo) is expressed by the equation -=Z>.i2tan'! which gives But the equations X>,i2 D,{i-) 2 cot? il il^i V' Qt give Cr ' D^u = cos " = cos 1 cos Z, j^ ^ 1 sin f cos ^ i>T U ^ D,{ud\\l)- and if ^1 is an arbitrary constant, Ds log V = ^<,log [u sin %) A z' = u sin^==«^ A" A v^ =^ u V sin 1 = ic" Dt I so that the area described by the projection of the radius vector upon ^ 335 — ihe 'plane of x y is projiGrtional to the square of the vclocify of the hodij. 577. The equation (33498) gives JJ„s^=^ sec 1 „^tan?^ Av _A I 2(0— .Q,,) ^ ■'• ?« M v' (it- — A'v') u V ?t- — 2 ^^ {SI — Sio) ' 578. If (5 is the angle which the radius rector makes with the axisj the preceding values give n , — J «^['-!+j(^^ '■)•-'] -^^ ^ ~ V «'-— 2 A^ (Si— si,y 9^ «-y M- — 2.1- (i2 — iio) When the forces are parallel these equations give A D,G I 2 (.Q — /2„) w V « (^ _ 2 A {.Q — iio) ' 579. . Upon the surface of revolution which is determined hy the equation B v=t(, in which B is an arbitrary constant, the value of 1 is by (3342^) constant, so that upon this surface tlie hraciiystochrone niaJces a constant angle with tJie meridian curve. In the case in which A=iB the brachystochrone becomes perpendicular to the meridian, and is a small circle, of which the plane is horizontal. Whatever is the value of B, the point at which v vanishes, — 336 — coincides with that at which ii vanishes, so that at the pole of this surface the velocity vanishes. Upon any oilier surface of revolution about the same axis, the incli- nation of the hr achy stochr one to the meridian arc is the same ivith the corresponding inclination upon the surface of equation (33592), ^'^ ^^^^ com- mon circle of intersection of these two surfaces. Hence the limit of the hrachystochrone upon a given surface of revolution is its circle of intersection ivith the surface of equation Av = u , and the hrachystochrone extends over that portion of the given surface, ivhich is exterior to the given surface, by which the limits are thus defined. 580. In the case of a heavy body, the surface of equation (33592) is a paraboloid of revolution. When the velocity of a heavy body upon any paraboloid of revolution, of tvhich the axis is vertical and directed dowmvards, is just sufficient to carry it to the vertex, the hrachystochrone maJces a constant angle ivith the meridian curve; but ivhen the velocity is too small to carry the body to the vertex, the hrachystochrone is a curve ivhich maJces an increasing angle ivith the meridian as it descends, and may sometimes become perpendicular to the meridian ; and luhen the velocity is more than sufficient to carry the body to the vertex of the paraboloid, the hrachystochrone is an infinite curve, ivhich is horizontal cd its highest point, and diminishes its angle ivith the meridian as it descends. If the equation of the paraboloid is l\^ z=:z 4: p ;S in. which the axis of 3 is the downward vertical, the equation (33408) becomes If .^0 is positive and p>iA^g, — oo7 — the substitution of • 2 ^'ff sin-" a = — -^ , 2p q ^= Sq tan^c?, ^ — p — q gives s z= J sec a {p — rj) ((p + Sin (p), in which the upper signs correspond to the case in which ]) is greater than + :?) cot a (sin (f + f|). The substitution of tan- w = , • 9 • <7 sni- 1 — P + 9 tjives iin a sj[p{p^rj)} When 2-0 is negative, in which case the condition (00630) is satisfied, the substitution of the equations (00T2--5) with the lower sign gives the corresponding value of (ooTj) for the arc measured from its upper limit, which corresponds to the vanishing of 9). When the substitution of cos- w = — f^ , . 9 , 7' + ~o snr I ^ — — jj — (J gives ;; = tan a y/ (^^) \^i 'H' — 3=^ 1// + cos ij' \ (cot- if -j- cos" /) i^g>,( ^,„.)]. -r/ \ p — (/ ■ /J ' p—q \ p—q When — •^o>7^ the sub.stitution of — 340 — 2 - + -\- Sm (/ ) . The substitution of tan-" L" = , • 2 • ^0 — q snr^^ -, — -, srives 582. In the case of a licavij hod// upon a vertical rujht cone, if the vertex of the cone is assumed as the origin, and if a is the angle which the side of the cone makes with the axis, A' q CO? a ^ =^ the angle which r makes with the axis upon the developed cone, the inclination to the meridian, the derivative of the arc and of ^ are Sin 1 = ^ '-^^ —^ , D.s jy^^ v /[2.,(;--n,)] r y/ [r- — 2 r^ (r — r^)] * When 2ro>ri, the substitution of snr2 'J. ) r, cot I tan w =: ^ r — r J r = i-Qsec^ Ig), — 342 — gives cos ^= cos ^ sec (i/' — /), s = Tq sin^ '/ (cosec iji — cosec 2 z) — r^ log (tan ^ iji cot ?") , ^ = — siii^"-^^ (sin / sin (p) -\- sin i '^\ (/> , in which the arc is measured from the cusps, at which point TJiis Inichjdochronc extemls to infinitfj from the cusjj tvitJiont ever Itecoming peiyendicuhw to the side of the cone. The greatest angle which it makes with the side is /, and at this point of least inclination to the side yi = i, r=2ro, (p=^^7i, 6=- — i-\- sin '/ ^ j ( ^ 71 ) . When the hrachjstochrone is defined hij the equation J <5 = ta„-y C;^ - 1) - Cot-y (1 - 1) , and the length of the arc, measured from the point of least inclina- tion to the side, is 5 =: r — 2 To + r, log {{^ — l)^ When ;■() is positive and the substitution of 9 r Sec^P* =-^^-^, ^ I r, Tan 3 1 an ii; r= + '- , — o4o gives s = j\ Tan [} Cosec !/' — rx log Tan ^- i/^ , in which the arc is measured upon each branch from the point at which it is horizontal and the upper sign belongs to the lower branch and the reverse. T/ie vpjjer branch is finite, ivhilc the lower branch is infinite, and the value of i/' extends on the upper branch from 2 [) to infinity, and on the lower branch from inhnity to zero. For the upper branch the substitution of sin i:^e~'^'\ r — ro = To sin i sin^ f/) , gives ^ = 2(l + sinz)[3^,9-^,(sin/,(p)]. Upon the lower branch the substitution of r — ^o = -. — . . ., , sill I i-th of the arc is • 5- = 2 y-i tan -^ t5 sec i (^ + 2 r^ log tan ( i tt + i (^ ) . When ^0 is negative, the substitution of ^ 9 ,j 2 ?n — 4 sin % Cosec" 5 = — — = i\ (i -j- ^i'l 0^ ^ — r, Cos/i' , V()^ ^1 — 344 — gives 6' = i'l Cot (■) Cosec If — Vi log Tan | i/' , in which the order of the signs and of the value of if is the same as in (3484) with reference to the branches. 27ie upper and finite branch of the t)r achy dochr one lies in this case upon the upper and inverted portion of the cone. The formula? (343ii, 343i6_i9), J^pply to this case, in which it must, however, be noticed that the sin i is negative. 583. When the solid of revolution upon ivhich the heavy hody moves, is the ellipsoid of which the equation is (z)+a)=i> the inclination to the meridian is determined by the equation The problem naturally divides itself into two cases. In the first case the velocity is more than sufficient to carry the hody to the highest point of tlie ellipsoid, tlie hracJiystocJirone is a continuous curve tvJdcJi is liorizontal at its liKjlicst and lowest limits, and tvJiich, always running round ttie ellipsoid, is most inclined to tlie meridian curve at tlie point In the second case, the velocity is not sufficient to carry the hody np to the highest point of the ellipsoid, and the hr achy stochr one is horizontal at its lowest point, hut has cusps for its nj)per points. In each of these cases the length of the arc can be found by means of elliptic functions. If in the first case — .'.2 are the coordinates of the upper and lower limits, or of the common intersections of the ellipsoid wdth the paraboloid of revolution of which the equation is n' = 2A'g(z-z,), — 345 — and if in the second case z^ refers to the intersection of the ellipsoid with the paraboloid, while — z^ is the coordinate of the intersection of this paraboloid, inverted at the horizontal plane of ux^ with the hyperboloid of revolution, of which the equation is (i)-(zj=i. the derivative of the arc is In the first case, ivhen the ellijosoid is p'olate, and the substitution of gives 2^ -r - Cosy— ~ ^\ "' , s = -f (^2 — ^i) ( Sin g) + 9) ) . When the ellipsoid is a sjphere, of which the radius is R, the hyperbola (SlSg) becomes equilateral, and the length of the arc, measured from the lowest point, is determined by the equation COS — R— z,-^z, • In the first case (344i7), the substitution of sintf;i COS« = -^ — -, 44 — 346 — gives, for the sphere, „ 2 cos"-^ 2 ^'i Tii /^^^ V^2 -\- 30S \pi s sin n>- ^'i Tji (^^^ V^2 + 30S li^j 5 \ 1^ 2 sin^ ^ Ti), (jp / cos Ti^2 + cos ipi s \ ~ i \ 1 _ cos \v7~ ' 2^/ "" sin 1/;. * \ 1 _j- cos V;^ ' 2^/ 4(,'0S"^ U'l :;)7) A'OSX/^-f-COSli'i 5 \ COS 1i>i COS Ip^ CTj; / * \ sin u^o * \ 1 — cos XI;., ' 2 ^/ sin if o * \2 E/ tan^-i] cosii;2 + cosT/;j \J (sin" 1/'^ cosec- ^ -\- sin- iX/i sec^ gl^) In the second case (34423), the substitution of 2 2 z — Zo — Zq (p =z ", sm I ^2 + 2^1 gives, for the sphere, „ Zj-\- R ,^^ /cosxl'o — COSl/»o \ Zi E (^ /cosxl'o COSti'i \ ^ E sin U)^ * V 1 — cos t!)2 ^ ^ Esinxp2 \ 1 + cos w.^ ' ' / ^1+^ \yn ( coi 'P2 — cos x/>o \ gj / i? (1 — cosi/»,) \1 i2cosii;,L 'V 1— cosi/»2 '^/ ^ \2, + i?cost//2 ' "/J I cos i/'o — cos 1'/., rrf= .. r 11 sin { sin q? -\ ^A ~ 9=: fp — cosec I tam~^J -7— -i ^ ., v . ' sin 1//2 ^ V (1 4- COS-* tan- q[.) In tlie case in which ^0= — ^, the brachystochrone is defined by the equation s ^^ 2^ ' tan^i/>2 584. In the case of a Iieavy body upon any surface whatever, it follows from (3203) that V- 2r/(z — z,) 6^i If, then, i\^^ is the normal to the bmchystochrone drawn in the tangent phane, and extended to meet the horizontal pLane from which the body must fall to acquire its velocity, the preceding equation gives i\;=(^ — 2-o)sec;^=^(^^, or ihe tangential radius of curvature of the hrachf/dochrone is twice the icmgential normal ivhicJi extends to the horizontal plane of evanescent velocity. This proposition is given by Jellett. 585. When the force is parallel to the axis and proportional to ihe distance from a plane ivhich is perpendicular to the axis, the surface of revolution of erpiaUon (33022) i-^ (^ti ellipsoid v:hen the force is attractive towards the plane, and it is an hgperholoid of two sheets when the force is repulsive from the plane. 586. ^Vhen the force is directed tovmrds a fixed point and propor- tional to the distance from the point, the surface of equation (33522) ^'^ ^'* ellipsoid if the force is attractive, hut if the force is repulsive, the surface may he an ellipsoid or it may he an hyperholoid of tvjo sheets. 587. When the force is directed towards a fixed point, and inversely proportional to the square of the distance from the point, the surface of revolution of equation (33522) is defined hy an equa- tion of the form = aQ—1). 588. Other conditions might be combined with that of the brachystochrone. Thus if the total length of the arc is given, the normal pfessui^e to the hrachystochrone is D^n = '^^^^=-{i+hv), in which h is an arbitrary constant, and is dependent, for its value. — 348 — upon the given length of the arc. This constant is generally infinite, when the brachystochrone is a straight line. 589. Under the condition of the preceding section, the equation of the hrachf/siochrone, in the case of § 569, referred to the coordinates of (33O17) is In the case of § 570, this equation gives 590. In the case of the parallel forces of § 571, (34728) P^(^s " M — oasmy When the force is constant, this equation gives so that when the curve has points of contrary flexure. 591. In the case of § 576, and ivith the condition of § 589, the equation of the hrachystochrone has the form l-\-bv The inclination of the curve to the meridian arc is therefore con- stant upon the surface of revolution, which is defined by the equation Bv=zu{l+l?v), and this surface has the same relation to other surfaces of revolution in — 349 — respect to tJie hrachjstochrone formed under the present conditions vrith those ivhich are indiccded for the surface of \ 579. In the case of a heavy body, the equation of this defining surface of revolution is ,2 2g(z-z,) = {-j^^ 592. If the condition is a mechanical one, such that the total expenditure of action, defined as in § 308, shall he given, the normal pres- sure to the hrachjstochrone is DM = - 1-2 1+5^2 l — bv-^ in which h is an arbitrary constant, and is dependent, for its value, upon the given expenditure of action. When this constant is in- finite, the normal pressure is equal and opposed to the centrifugal force. It is apparent, from the preceding equation, that under the action of finite forces, this brachystochrone cannot be a continuous curve, in one portion of which the direction of the normal pressure coincides with that of the centrifugal force, and is opposed to it in another portion. 593. Under the condition of the preceding section, the equation of the hrachystochrone, in the case o/" § 569, referred to the coordinates of (33O17) is /2 — i2o _ —(Py^ [\J^2b{^ — fi,)j \pj' In the case of § 570, this equation gives ^ 1 — 2ba(r^ — r^) ' Pisfoi 594. In the case of the parallel forces of § 571, (3492,i) gives — sm^ ^, . [I -{-2b (Si — Si,) J — 350 — When the force is constant, tliis equation gives 595. Ill the case o/" § 576 and tvith the condition of § 592, the 'equation of the J)vach>jdochrone has the form The inclination of the curve to the meridian arc is, therefore, constant upon the surface of revolution, which is defined by the equation and this surface involves^ for the present case, the iiroperties of the defining surface of § 579. In the case of a heavy body, the equation of this defining surface of revolution is 2 i?V (^ - ^«) = «^ [1 + 2 % (2 - z,)-\\ 596. The hr achy stoclir one in a medium of constant resistance is entitled to special consideration. In this case, it is convenient to introduce the length of the arc as the independent variable. The equation of motion along the curve is v^=.2n — 2Jcs, in which Jc is the constant of resistance. This equation must be combined with the equation If h ,"i and ^ ,"' are the respective multipliers of these equations in — 351 — the method of variations, the hrach/jstoclwonc is defined h/j the differential equations f'i = ^, and ])y the following expression of the normal pressure directed in the opposite way to the centrifugal force Z>^ X2 sin V -\~ D^S2 cos v =. - — . When /c vanishes, the value of ^t is 1 and, therefore, the value of ii is the negative of the reciprocal of the ex- jpression which is oUained for v ivhcn there is no resistine/ medium, and ivhich is independent of the magnitude of the fixed force. 597. When the force is directed toivards a fixed centre, the nota- tion of § 569 gives by (ooOjg) for the value of /x, 598. When the forces are parallel, the equation (oolgi) gives II in the form ' COS V 599. From the preceding ecpiations, the crpiation of the hrachgs- tochrone of a heavy hody in a medium of constant resistance has the form R sin V ^ ~ fl— /icos(»' — ro)J ' in wdiich R, h, and Vq are arbitrary constants. — 352 — 600. In a medimn of ivhich the Imv of resistance is expressed as a given function of the velocity^ the derivative equation of mo- tion is in which F is a given function of v. The differential equations, by which the brachystochrone is defined, become, if I jU, and ji/j are the multipHers of (35O29) and (3524), D, [ji sin v) = D^ £1 D, ^1 , — D, {^ sin v) = nj2 n.pi, The reduction of these equations gives and the expression of the normal pressure to the brachystochrone becomes D^ 12 sm V -\-D^ll cos v = -^— = —^r-. = —^ — rri^ — fix Vv^ -|~ ^^ QH-^v^D^ V — q' 601. When the forces are parallel to the axis of z, the equations (3529) and (352i7) give a ^ sinv' ir a \ ' • sm V V — 353 — G02, These equations give for the h-achjstoclirone of a heavy hodj ill a rcsistiuf/ medium, by wliicli V is determined in terms of v. The substitution of this value of V in the equation t* D^. V j^ - — — = — a cos )' — y, gives the equation of the hrachystochroiie in terms of q awl r. The pre- ceding formula} include the results obtained by Jellett in his inves- tigation of this particular case. When V is inversely proportional to the velocity, the equation of the brachystochrone may assume the form 2 A [A cos 2 (v — a)-^kjsm 2(v — a) Q = m -j- g cos r [/( cos 2 (r — «) -h ^'J When V is proportional to the square of the velocity and has the form the equation of the brachystochrone is derived from the elimination of V between the equations cos(v — a) = g cos a f/ fin r cos « /gcosv j\ [i')qco>a o / \l/.7Cosr , ,\ k7i;i^X—' ''■) = [-^' ocos(v-a)J(-^ + /-). G03. In these cases of the brachystochrone in a resisting medium, it is apparent that the condition (3296) is usually violated, and that Euler, consequently, erred in extending this proposition to the case of the resisting medium. 40 — 354 — 604. The determination of the form of the curve constitutes the principal feature of the general problem of the brachvstochrone. But the nature of the curve may be given, and the problem is then reduced to one of maxima and minima, in which the various param- eters of the curve are to be determined. Euler has shown that there is a peculiar anah^tic difficulty in some problems of this class. A single example will illustrate this species of inquiry. Let the given curve be the circumference of a circle, of which the plane is vertical, and let the ball start from a state of rest at the upper point. If, .then, 2 a is the angle which the line, joining the two points, makes with the horizontal line, and if 2 i is the angle which the radius drawn to the upper point makes with the vertical, the equation for determining i is sec / [f ^ ( 1 jt) — I, ( 2 « — /)] — [cot 2(i — a)-\- cos /] l-^,(in)-7f,{2a-i)] + '-^^ cos 2 a I sin 2 («" — a) 0. sin z « THE nOLOCIIKOXE. 605. A curve, in which the time of descent along a given arc, is a given function of the arc, or of its defining elements may be called a holoehrone. 006. The problem of the holoehrone becomes simple, tvhen the forces are fixed, and the tunc of descent is irroportional to a given 'power of the arc. Thus, if the time of descent is expressed by T,= As'\ in which s is the leng-th of the arc. Let ±(1 — 355 — in wliijli the upper sign corresponds to tlie case, in wliicb n is less than unity, and the lower to that in \Yhich n exceeds luiity. The force alonti; the curve is When the force along the curve is 2 A.,s s ' GOT. When the force is that of gravity, the equation of the holochrone of the preceding problem assumes the form g sin T r=: — B b^~'~'\ 608. If the iime of descent admits of heing devetoped according to integral ascending poivers of s, the developed expressions of S and 12^ are obtained from the formulce in which the successive terms of P^ are obtained from the equations represented by W2L,=X['=-9r:fc)]- 1 The second member of this equation is to be developed in form precisely as if f were the symbol of derivation, and in the result there must be substituted for P, = o ^"^^^ f^''^ J-\=.q, the values — 356 — 609. W/mi the forces are fixed, and the time of descent is a given function of the iniiial value of the potential, the problem of the holochrone can be solved by the method applied by Abel to the case of a heavy body. If A is the final value of the potential, in which the arbitrary constant is determined so that the potential may vanish with the velocity, the time of transit expressed as a function of A, assumes the form T — i- r ^^' ■^^-' s|•^J^^J{A — ^y T\\Q integral, relatively to A of the product of this expression, multiplied by nsl{il—Ay is But the notation rh=J^{—\ogxY-\ with the familiar equation 1 J^ x«-^ ra r(i— k) gives, by a ready reduction rv 1 C _^^LZ!LJ\ — — rnril ;A— ^^° ;Note. — The notation (SoG^o) is substituted for that of (91,,4), which was unwisely introduced instead of the usual form, which is here restored. €)■)( If the product of tins equation multiplied by cc ff («) is in- tegrated relatively to a, and if the function /^ of x is defined by the equation r(9(«)2-)=/„ t/ a so that U a the integral ffives which, when gives by (ooSie) n Ja \/{Si — S — * The general relations between s and 12 complete the solution, and indicate the form of coordinates in which the solution should be finally exhibited, 610. If the forces are parallel to the axis of z, 12 is a function of s, and the elimination of z between (ooTis) ^'^^'^ the equation cos^ = A-5, gives this holochrone expressed in terms of the length and direction of the arc. Gil. If the forces are directed towards a fixed point, which is assumed to be the origin of coordinates, the elimination of ;• be- tween (SSTis) ^^^ cos ' = D,s, gives this holochrone expressed in terms of the length of the arc and its inclination to the radius vector. — 358 — G12. If T^, developed according to powers of A, is expressed by it is evident that -n/^-4^;^>-] 613. An interesting case of this potcniial lioloclirone is olttained, ivlien the hodij is supposed to approach the point of maximum potential along a given curve, and the required curve is to he such that the whole time of oscillation shall he a given function of the maximum potential. If 5i denotes the given arc, the time of oscillation has the form rp _J_ f D^ (^ + ^0 . so that, by the process of § G09, u In order that the two curves may be continuous, the direction of the given curve must coincide with that of the level surface at the point of maximum potential. But this direction may be given by an infinitesimal bend at the extremity of the curve, so that this is not a practical limitation of the problem. 614. If the given time of oscillation is constant, the equation (oSSig) assumes the form B{s + s,) 2 — r) f and the compouml curve hccomes a peculiar species of tantochrone, which was investigated by Euler in the case of heavy bodies. 615. When the forces are not ivhollg fixed hut mag depend upon — 359 — the vehcitf/, the 'problem of the holochrone hccomes, to a certain extent, indeterminate. For, if TF=zO, is an assumed equation between 5, t and v, such that t and s vanish together, but when v vanishes, the resulting equation between 6- and t assumes a given form corresponding to the given condition of the holochrone, the derivative of this equation gives, for the expression of the force along the curve, R from which the time is to be eliminated by means of the assumed equation. 61 G. In most problems, in which the forces are dependent upon the velocity, the form of R is not unlimited, but is usually so re- stricted that in ti'hich Eg is a function of s and represents the action of the fixed forces, tuhile E^, is a function of v and represents the resistances, to which the body is suhjcct. In this form of the jDroblem, geometers have not made much progress towards its solution, although the case of the tautochrone, exhibited in this aspect, has been the occasion of much discussion and many difficult memoirs. 617. If the equation (SSOg) solved with reference to /, ac- quires the form the expression for JR is l — v n. T. ,. B -D.. Z. Avliich is essentially identical with L.vciRAMiE's most gn/cral foj-mula in the case of the tautochrone. — 360 — 618. If the equation (SSOg), solved with reference to v, ac- quires the form the expression for R is Avhich formula comprises Laplace's general form of solving the iauto- chrone. 619. If the equation (SSOg), solved with reference to s, ac- quires the form the expression for R is p V — A S,., t 620. When the equation (SSOg) is presented in the form in which T, >S', and V are respectively functions of ^, s, and v, the A'alue of 7t is R = — A V But D^T h a function of t and, therefore, of S -\- V; it may, indeed, be any arbitrar}^ function of >S' -|- V, so that if ijj denotes this arbitrary function, R becomes 621. When, in the preceding section, aS' is changed into — log aS' and V= log 2?, — 3t)l — the value of H may be presented in the form which is the same with a familiar formula of Lagrange for the case of the iaidochrone. 622. The cases, in which the formula (SGlg) assumes the form (359i5) ^I'G easily investigated. For this purpose let V and the derivatives of (oGlg) give 7>„ 7? = A / + 2 1' A log S^D,. i?,, D,D,R = —^^Dli^ 2z SDs A log S= ; whence i)2;^ = 2>S''AAlog>S'=2«, in which a is any constant. Hence 7, = az'^-\^lz-\-c, in which h and e are constants introduced by integration. The value of R is, then, R^c,SJ^hv + (a + A 'S^) I' ; so that, if /^ and II arc constants, the final values of S and R are R^eS + hv + hi^-, 46 OUJ — and this formula of Lagrange is restricted to the resisting mediimi, in vjJiich the resistance has the form a -\-1j V -\- h v^ which was first remarked by Fontaine. The form of T, in this case, may be derived from the equations T c ' — z, — I), T=^ = az + lj + '-=t> + ec''+ac-'^; which trive 2 e av S -\- be S' -\-h a ir bv S-\-eS'^^av' y^h.^ilLl V vanishes this equation becomes sJ[1ea)^Q^\{r.—t)s){;iea — t?)'\ = h, so that the interval r — /is independent of the length of the arc, and the curve is a tautochrone if % is also independent of s, which is the case when S vanishes with 6', that is, when h This condition is always observed, if the direction of the curve coincides witli that of the level surface at its termination, so that in every case, this holochrone is cssentiaHij tautochronous. 623. If, instead of (3592.r,) we suppose and if (/' denotes an arbitrary function, the value of 11 has the form l^'l} ■*■ -«. V OP o OUO When in "svhicli aS' and iS\ are functions of s, and V is a function of v, the valne of II becomes * = ST- [''■ f-^' ^"+ *■) - '• (*' f'+ '^' )] ' which inchides Lagrange's formula. Forms of this kind may be indefinitely multiplied, without diminishing the difhculty of obtain- ing such as are new and not included in the iuvestigations of § 622. 624. A curious case of the holochrone is introduced, when the form of H is in which S\ is a function of s. The only case of (36I3), which can assume this form is easily proved to be that of (361gi) when jS is left undetermined. If, then, the factor of r^, diminished by a constant, is inversely proportional to the radius of curvature, t/ie form of the t^esisiance, by including in it part of the term e S, is that of (0623) increased !)>/ a term 'proportional to the friction upon the curve. If the fixed force, in this case, is that of gravity, and the axis of s is vertical, and if v is the inclination of the radius of curvature to the axis of z, the first and last terms of R give, if k is the con- stant of friction, ^ q 9^\x\v -\-1c g cos v O = — ' , e a + D, aS'= a — - = — ( A 4- - ) >S , 1 a e — hg^mv — hhg cos v ^ Q (1 — F) g cos V ' — 3G4 — so that the curve determined hi/ (oGlgo) isjncluded in this form. This is a generalizi to the cycloid. is a generalization of Bertrand's similar investigation with regard THE TACIIYTROPE. 625. A curve in which the law of the velocity is given may be called a tach/jtropc. 626. When the law of the velocity is given in an equation Jjctiveen the velocity^ the space., and the time, the formulcv o/* §615 are directly applicable to the complete solution of tJie prohlem ; and all the subsequent transformations of these formulce may be applied to the present case. 627. When the time is not involved in the equation (oSQs), but the portion R^ of the force R is given, the other portion R, is determined by the equation *« JJ ]Y ^''5 from which v is to be eliminated by the given equation (ooOgj). EuLER has solved various cases of this tachytrope, 628. One of the simple examples, solved by Euler, is when, in the case of a heavy body, ^„ = — /t?''% and the velocity is to depend upon the arc in the same form as if the body descended in a vacuum upon an inclined straight line, so that the equation (3593) iicquires the form V^ :=. hs, whence y sin V = R^=z^h-}- k (h s)'"". 3G5 — When this equation becomes m g sin V ^=1 ^ h -\- Jc hs, or the required tachjtrope is a cycloid. G29. Another simple and interesting example of this prol^lem was proposed by Klingstierna and solved by Clairaut. It is that of a heavy body in a medium, of which the resistance is propor- tional to the square of the velocity, approaching the origin with a velocity equal to that which it would have acquired by falling in the same medium through a height equal to the distance of the body from the origin measured upon the curve. In this case - whence the equation of the tachytrope is of which the integral is 630. A simple example of the problem of § 627 is that in which the velocity is uniforui. In this case 7?, =: — B, = a constant = A -- , so that ill the case of a licavij hodij this tachjtrope is a straight line ; in tlmt of a constant force directed towards a fixed point, it is a loga- rithmic spiral ; and in every case the sine of the angle, at which it inter- sects each level surface, is inversehj proportional to the fixed force u'hich acts at the iioint of intersecticn. — 366 — 631. When the given forces are parallel to the axis of ^, and the given equation (ooOg) is expressed in terms of v and 0, the equation of the tachytrope is {D, n sin V -\- B,) D,. W-\- A Wv sin v = 0, from which v is eliminated by means of the given equation. Euler has solved several cases of this tachytrope. 632, If, in this case, the curve is to be such, that the velocity shall have a constant ratio to that which it would have acquired in a vacuum, the equation (3665) assumes the form D, 12 sin V z=z — 1 — « If the resistance is proportional to the square of the velocity, so that li,. has the form the equation of the tachytrope is sin V A log (/2 -f- //) = ^. 633. When ihe (jivcn forces are directed towards the origin, and the given equation (3593) is expressed in terms of v and r, the equation of the tachgirope, in a medium of given resistance is {D, n cos : + i?„) A- W-\- D, Wv COS : = from which v is eliminated by means of the given equation. 634. If, in this case, the curve is to be such that the velocity shall have a constant ratio to that which it would have acquired in a vacuum, the equation (36624) assumes the form A-^^cos; = — -^. 1 — a — 867 — If the resistance has the form (36Giq), the equation of tlie tachytrope is 2 a k cos: A log (/> + //) I— a' Goo. W/icii the laiv of the vclocibj, in a medhun of Jrnown rcsid- ance, is given in a given direction, such for instance as that of the axis of X, and so given that t'cos^= W,^^, in Avhich W, j. is a given function of s and x, the equation of the tachytrojDe is derived from the equation {D, n -]- li^) cos ^ — ^^ sin ^ = ^' A W,^ ^ + v cos ^ D^ W,_ ^ ; from Avhich v is eUminated by the given equation. 036. When the velocity in the given direction is uniform, these equations become V cos % = a , a- sin ^ 637. When the given force is that of gravitg, and (i is the in- clination of the given line to the vertical, the eqliatiou of this tachytrope hcco)ncs (5rcos(p,' — ;:) + i?,) cos'' This problem is solved by Euler in the case in which the given direction is horizontal and in that in which it is vertical. A special solution is obtained upon the hypothesis of a constant velocity ; in this case, the tachytrope is a straight line determined by the con- dition ^cosGi — :;) + i?„ = o. — 368 — 638. When there is no resisting medium, the equation (oGTsi) ({f the tachytrope becomes ^ g cos^ tc cos {^ — ly When the hne is horizontal and the equation becomes a" ^ g cos^ % so that the tachjtrope of this case is a iiarahola. When the line is vertical /^ = 0, and the equation becomes a sin \. (> = n? SO that the tachjtrope of this case is the evolutc of the ijarahola. With the notation 2 dr the equation (3683), expressed in rectangular coordinates is 11)\l{x-^y cot /i) — t)x = lQoi (-i log [cot l^>-\-b^{x -\-y cot f:?)] . 639. If the resistance is proportional to the velocity, so that and if the direction of the line in which the velocity is given is such that g cos /•> ■=^ka, the equation of the tachytropc of a heavy J)ody is X sin {i — y cos [■) z=z — c a , 369 THE TACHYSTOTKOPE. 640. The curve on which the final velocity in a given resist- ing medium is a maximum, may be called a tachjdotropc. 641. In a medium in which ihe law of resistance is expressed as it is in § 600, the notation of that section gives for the differential equations of the tachystotrope D^ (u sin v) = D^ 11 D, //j, — D, (a cos J') = D, £2 D, //j , V D,Ui = III D^ V. The reduction of these equations gives D,^i = n,S2I),^,i = I),{u,V), ^ = Hi V, and the expression of the normal pressure to the tachj^stotrope be- comes V cAiogK' 642. In the case in which the law of the resistance is ex- pressed by the formula the normal pressure becomes D 12 = — P in () ' so that the normal press^ire has a constant ratio to the centrifugal force, wdiicli result was obtained by Euler in the case of a heavy body. 47 — 370 — 643. When the resistance is constant, the tachjstotrope is a straight line. 644. When the forces are imrallcl to the axis of z, the equa- tions (3699) and (oSDie) give fi = u^ V= - a sin >'' 645. The equation of the tachjstotrope of a heavy hoJij is ob- tained, therefore, Inj the elimination of v hetivccn the equations V= ^" b sin V — a cos v ' V D^, V a a — = — q COS V — -j—. — ^^ . (> "^ b?,mv — acosv 646. When V has the form (3692,i), the equation of the tachy- stotrope of a heavy hody is a -J—. -T, = Jr- (m a o sin v\ {b^va.v — acosi')' \ t/ >> j THE BARYTROPE AND THE TAUTOBARYD. 647. The curve, in which the law of pressure is given, may be called a harytrope, and that bar^^trope, in which the pressure is everywhere the same, may be called a tautoharyd. 648. When the pressure is a given function of the arc, which may be denoted by S, its equivalent expression, if F is the fixed force which acts in the direction /, is - — i^sin{==>S'; and the differential equation of the harytrope is 2R = D, [*) (^'4- i^cos,0] = 2 ^sin;;+ 2 E,. — ot L — 649. Iji ilie case of a hcavn hodf/, if the axis of z is vertical, the differential equation of this havjjtrope hecoincs from which v may be ehminated by means of the equation ^'^ :=! ct [S -\- g sin v) . In this case, the differential ecjiiation of the tautoharyd is (a -{-(/ cos v) D, = 3^ sin v -\-2B^. 650. ^yhcn the resistance is constant, the equation of the harf/trope of § G48 is i> {S + Fco.p = 2 [n + //) +2 .7?„. In the case of the heavy lody, this equation becomes ^^■i?,5+^cosi/7?,. = 2y.- + 2//+257?„j and that of the taidol}aryd is [a -\-g cos vj"^^ () =. A \_g -f- a cos v -]- sin v \J (y^ — ^^^'YW if A is an arbitrary constant, and But if the equation of the tautohargd is lo^' K ( ^^ + y COS V Y = -y—i ^ cos^ ^J ~ . O'-Q When there is no resistance, the tautoJ)ar>/d of the heavy hocly is defined hij the equation _ A 651. When aS' vanishes, there is no pressure against the bary- trope, and this curve is that on which the body moves freely. Thus the equation of the barjtrope of the heavy body becomes, under this condition, A ^^ ~ (cos rf ' tvMch is that of a parabola. 652. When the curve of the harytrope is f/iven, the eqiicdions (37O27) and (STOgi), determine the laiv of the fixed force when that of the re- sistance is hioivn, or, reciprocally , that of the resistance, lehen the fixed force is hnown. 653. When the forces are parallel to the axis of z, the equation (37O31) becomes * \ s / I COS- ;' lohich is appliccd)le ivhen the curve is given. When there is no resistance, this equation gives Fo cos-^ V = — / [cos^ V n, {S'q)-] = — ^ [cos^ )/ n^ {S())'] . 654. In the case of pa?rdlel forces, when the tautobaryd is a circle, and there is no resistance, the fixed force has the form F=: Q COS' V m which b and F must vanish, if v can become a right angle. When the fixed force is that of gravity, and the tantobaryd is a circle, the expression of the resistance is J!. = -lff{^-a) 655. In the case of jmrallel forces, when the tantobaryd is a cycloid of which the base makes an ano-le a with the direction of the parallel forces, and when there is no resistance, the equation of the cycloid being ^ r=2 ^ sin (j^ — a), the expression of the force is ^ «sin(r — «) 4- ^ a sin (3 * — «) -j- ^ a sin (r -]- «) -|- 5 2 sin (r — a) cos' v When h vanishes and a is a right angle, this expression is re- duced to F=^ I a coseci', which coincides with Euler's solution of this example. THE STNCnROXE. 656. The surface or curve which is the locus, at any instant, of all the bodies which start simultaneously from a given point w^ith a given velocity, and move upon paths which are related by a given law, is called a si/nchrone, and the given starting point may be called its dynamic pole. This class of loci was first discussed by John Bernoulli. 657. If an integral of the motion of the body along one of the paths to the synchrone is obtained in the form TF=0, — 374 — in which W is a function of the time, of the arc of the path, and of the parameters by which the relationship of the paths is expressed ; this equation is the required equation of the synclirone^ if the time is as- sumed to be constant ; and it is referred to the system of coordinates, con- sisting of the described arc and the given parameters. 658. If the only force is that of a resisting medium, and if the form of the path is given, and also the position of the dynamic pole upon it, but not its direction in space, the synchrone is obviously the surface of a sphere, of ivhich the dynamic pole is the centre. 659. If the body moves, without external force and without resistance, upon a straight line, which rotates uniformly about a given axis passing through the dynamic pole, the synchrone is a surface of revolution about the same axis, and it is defined by the polar equation (SSOso) or (25I3) ivlien p vanishes and t is constant. 660. When the fixed forces arc directed toivards a point, or ivhen they are p)arallel, the synchrone of bodies moving upon straight lines, is a surface of revolution, of which the axis is the line of action tuhich passes through the dynamic pole. 661. In the rectilinear motion of a heavy body, it is obvious from (255i3), that the 2^olar equation of the synchrone has the form r = a cos l-\-b, ivhich becomes a sphere, ivhen b vanishes, that is, ivhen the initial velocity vanishes. 662. In the rectilinear motion of a heavy body through a medium, of ivhich the resistance is proportional to the square of the velocity, the polar equation of the synchrone has the form, Ac''=Cos{Bcos^:). — 375 — THE SYXTACIITD, GG3. The surface or curve ^vliicli is the locus of all the poiuts, at which bodies have the same velocity, when they move from a given point, Avith a given velocity, upon paths which are related by a given law, may be called a Sf/ntachi/d. 664. If an integral of the motion of a body along one of the paths which proceed to the syntachj'd is obtained in the form W= 0, in which W is a function of the velocity, of the described arc, and of the parameters, this equation is that of the syntachyd in the same form of coordinates with those in which the sj^nchrone of § 657 is expressed. 665. In the case of §658, the syntachjd coincides ivitli the sijn- chrone. 666. In the cases of §§ 659 and 660, the syntachyd is a surface of revolution about the same axis iviih the synchrone. 667. When the action is exclusively that of fixed forces, the syn- tachyd is a level surface. 668. ^Yhen a heavy body moves upon a straight line, on which there is a constant friction, and through a medium of ivhich the resistance is proportional to the square of the velocity, the equation of the syntachyd is in which the notation of § 5] 5 is adopted, A and B are constants and a = g tan a . 669. When a heavy body moves vpon a straight line, on which the friction is constant and through a medium of ivhich the resistance is 'proportional to the vclocit/j, the cqiuition of the srjntachjd has the form log iA - cos (. + .)] = i? - ^^^-^^ . 670. When the hody moves upon a line on vhich the friction is constant and through a medium of tvhich the resistance is proportional to the square of the velocdi/, the equation of the sf/ntachf/d, expressed in the form of coordincdes of § 657, is which coincides with Jacobi's investigation of this case of motion. A POINT MOVING UPON A FIXED SURFACE. 671. Among the various forms, in which tlie motion of a point upon a fixed surface, with fixed forces, can be discussed, that of tlie principle of least action is here selected. In this case, therefore, the whole amount of action, denoted by is to be a minimum. If, then, the equation of the surface is X=:0, if rectangular coordinates are adopted, if pi is the multiplier of the preceding equation of the surface, and ^a that of the conditional equation the equation of the path of the body, with reference to either axis, is D. V + a, D,L — D, ifi .1-0 = . — 377 — The sum of these three equations, multiplied respectively by x\ y, and /, is or V ^ a. Whence 672. If the tangent plane to the given surface is assumed, at each instant, to be that of .r//, and if the axis of jj is taken normal to the path of the body, the preceding equation becomes, if Oj de- notes the radius of curvature of the projection of the path upon the tangent plane, — D 11- so that the ccntnfvgal force of the lody in the direction of the surface to vMcli it is restricted is equal to the normal irressiire upon the path in the direction of the tangent plane. 673. ^Vhcn the direction of the force is normal to the surface, tvhich is the case ivith the level surface, or tvhen there is no force, the path of the hody is the shortest line tvhich can he drawn upon the surface, and coincides ivith the hrachystochrone. 674. When the velocity is constant, the erpiation (377]3) exj^resses the condition that the hody may move upon the intersection of a level surface ivith the given surface. In this case o^ is the radius of curvature of this intersection, and Dy 12 is the whole force in the direction of the tangent plane to the surface. 675. When the velocity is a given function of the parameter of the level surfoce, the equation (377i3), with the notation of the pre- ceding section, expresses the equation of a surface over which the body moves upon the intersection of this surface with the level surface, 48 — 37S — 676. When the force is directed toivards the origin^ and the given surface is a plane passing through the axis, the equatiou (oTTi.s), com- bined with (316i8), gives in the notation of § 569 D,£l 2 2D,p SI — i(iu (I sill ', p ' of which the integral is (2 o = ^ z= i r = -^ Dt s^. Whence, if (p is the angle which r makes with the axis, ir'I),^,=pl But i t^dcf is the elementary area described by the radius vector in the instant di, and it, therefore, follows that the area described lu the radius vector is proportional to the time. The equation (STSu), combined with that of living forces, gives i;,r — rv/ [(2 r- {Si — Sl,)—4.pW ' Whence ^ ■" ' rsJl2i^{U — Sl,) — lp\]' -I which is the polar equation of the path of the hodg. That this equation can be obtained by integration by quadratures, is a simple case of the principle of the last multiplier. 677. When the potential of the force has the form i'- = -^, and the initial velocity is such that 12 — — 379 — or l^ r=z 2 /2 = — the polar equation of the path of the hodij is plr"-'=\f{^a) sin [(;^ — 1 ) (9 — «)], which was given by Riccati. K the law of the force is that of gravitation^ and the path is a ^^wr^Jo/^ cf vMch the oricjin is the focus. If the attractive force is inversely proportional to the cuhe of the radius vector, and the pcdh is a logarithmic spiral, which was proved by Newtox. If the attractive force is inversely proportional to the fourth power of the radius vector^ and the path is the epicycloid formed hy the exterior rotation of a circle upon an equal circle, which was proved by Stader. If n = 2, the attractive force is inversely p>roportional to the fifth iwwer of the radius v^tor, and the path is the circumference of a circle, which was proved by Newton. If the attractive force is inversely proportional to the sixth power of the radius vector, and the path may he called the trifolia of Stader, by whom it was investigated. — 380 If n = o. the attractive force is invcrscl// proportional to the seventh poivcr of the radius vector, and the path is the Icmniscate of James Bernoulli^ which was proved by Stader. If ' n=^ — 1 , the repulsive force is proportional to the radius vector, and the fath is an equilateral hjperltola. When r becomes infinite when (^ — a) vanishes, which was remarked by Stader. 678. When the vahies of £2, S2q and ^h ^i'G such that, if II is an integral function of an integral root of r, B = s/{_2r^O-n,)-4.pr], the expression of cp in (87822) admits of integration. For if the integral root of r is denoted by ri= \Jr, and if the notation of the residual calculus is adopted, the equa- tion (37822) becomes mi . log(Vr— ri) ^ = 2»^'?i;;;^>=^»^'^i."&i 679. An example of the preceding section occurs, when m is unity and 11 =z a r^ ~{- ^ i' -\- c , — 381 — •which corresponds to n,=z — hV- — ac, and an attractive force of the form — a" r — ab-\---v,-\ -^^-^ . In this case the value of (p is •^ e ^R e\J{A:ae — b') sj {Aae — b'y = — log; ^ H , ,,., . — - 1 an^ ^J -j-jr, — -, — r When h vanishes, these expressions become n 1 ,,2 .2 _i ^'' "Mt'i the attractive force is and the equation of the path is « + ^= -i;(f-«). When e vanishes, the expressions become n — 1 7.2 the repulsive force is 2 1 7 ^P* a r-\- ao — — r , — 382 — and tlie equation of the path is When 1? — 4 (^ e vanishes, the equation of the path is :, (ffi (Z) = — - — W ( 2 « 4- -) . 680. Another example of §678 occurs when r, = ar + i+l, which corresponds to and an attractive force of the form «5 , &- + 2 « e + 4 joj . 3 i e , 2 fi2 The equation of the path is ^V(^^-4..)Tan[ N^^^^-^-) (y-.)]. When « vanishes, the value of 11^ vanishes, the attractive force is P-^ipi . She . 2e^ ^.i ? and the equation of the path is hg{br + e) = ~{(p — a). When P — iae vanishes, the equation of the path is 9 ar~\-b=z a — fjp — 383 — 681. Another example of §678 occurs when in which case and the equation of the path is nB i(« — 9) = log(l + ^r "'), 2 mp 682. TJie forms, in vMch (37822) admits of explicit integration iiitliout any special determination of S2q and p, are included in the f/cn- eral expression in v.'liich h is two, or the negative of unity, so that SI only consists of two terms, of tvhich one is b and the general fonn of the central force consists,, therefore, of two terms of ivhich one is inversely proportional to the cube of the radius vector, and the other may he either xlirectly propoiiional to the radius vector, or in- versely proportional to the square of the radius vector. 683. In general, it is apparent that the addition of a term to the central force, which is inversely proportional to the cube of the radius vector, does not augment the difficulty of determining the path of the body. In any ecpadion of a path of a hody described under the action of central forces, which is expressed by the elements (p — a,Y and t,and tvhich may also involve the constant pi, the multi- plication of the angle cp — a, and of p^ by the factor B Ai-4)' — 384 — gives the equation of the ixdh, when the central force is increased hj the term G84. When there is no force the path is a straight line, so that ivhen the central force is inversely projwrtional to the cube of the radius vector, the polar equation of the path is r cos [B [ip — a)']-=: B plJ If the force is repidsive, B exceeds unity, the path is convex to the origin, and its convexity increases with the increase of the repulsive force until it terminates in a straight line. If the force is attractive, and B"^ positive, it is less than unity, the path is concave to the origin but of infinite extent, and the concavity increases with the increase of the attractive force until it terminates in the reciprocal spiral of Archimedes. If the force is attractive, B'^ negative and I2q positive, the equation of the path is r Cos \B (c/, _ «) y/ - 1] = ^;>? \/^, so that the greatest distance of the path from the origin is limited, and the path is a spiral about the origin in which it terminates, at each extremity, through infinitely compressed coils. If the force is attractive, and B"" and /ig negative, the equation of the path is r Sin {B (r/) _ a) y' — 1] = B p\ y/— , so that the curve extends to an infinite distance from the orig-in at one extremity, and terminates in an infinitely condensed coil about the origin at the other extremity. In these three cases, the formula — 385 — for the time which corresponds to (8849) is t^ — -^iVii\\_B (ip — a)], the formula for (3842o) is and that for (08427) '^^ t = ?J^=^ Coi[_B [cf — a) si — r]. This law of central force has been discussed by several geometers, and, with peculiar regard to the special cases of the problem, by Stader, whose results coincide substantially with those of this section. 685. When the central force is ]j'^opo7iion(d to the radius vector, the path is a conic section of ivhich the centre is at the origin. It is an ellipse, if the force is attractive, and an hyperhola, if the force is repidsive. In the case of the ellipse, if a point w^ere to start from the ex- tremity of the major axis at the same instant with the body, and move upon the circumference of which this axis is the diameter, with such an uniform velocity as to complete its circuit synchro- nously Avith the body, the body and the point are always upon a straight line which is perpendicular to the major axis. For dif- ferent ellipses, the time of description is proportional to the square root of the area. In the case of the hyperbola, if a catenary is drawn through the extremity of the transverse axis, in such a position that this axis is the direction of gravity, while its ex- tremity is the lowest point of the catenary, and of such a mag- nitude that the radius of curvature of the catenary at this point is equal to the semi-transverse axis, and if a body starts upon the 49 — 386 — catenary simultaneously with the given body, and proceeds in such a way as to recede uniformly from the transverse axis with a velocity equal to that of the given body at its nearest approach to the origin, the line which joins the two bodies will always re- main perpendicular to the transverse axis of the hyperbola. 686. W/icu in addition to the term, which is ^J^oporiional to the radius vector, the central force has a term inversely 'proportional to the cube of the radius vector, the path can he derived from the preceding section hy the principle of § 683. When the term which is proportional to the radius vector is attractive and expressed by a r, the polar equation of the curve is '-^^ + ^-^ = v/[^^^^^o-4^^^^;4]cos[2i?(c^-«)] = y/ [ 02 _ 4 « i52^4] Cos [2 i? (9) — c^) v^ — 1] r= v/ [4 « i?2;4 — '^-^a Sin [2 i? (9) — 6.) V/ — 1] . When a is positive, therefore, the path does not extend to infinity, although when B^ is negative it is compressed at each extremity into an infinite coil. But when a is negative, the term propor- tional to the radius vector is repulsive, and the curve extends to infinity if B^ is positive ; but if B^ is negative the curve is limited if lio i*^ negative, or it may necessarily extend to infinity if il^ is positive. In the special case of tan (2 n B71) p\ , Yb — H.v— «? -^0 the curve is asymptotic to itself 687. When tlie central force is inversely proportional Jo the square — 387 — of the radius vector ivliich is the law of gravitation, the yath is a conic section, of ivhich the origin is the focus. When the force is attractive, the imth is an ellipse if il^ is positive, a paralola if S2q vanishes, and it is that branch of the hgperhola ivhich contains the focus, if S2q is negative. Bui when the force is repidsive, the path is that branch of the hgperbola ivhich does not contain the focus. The farther consideration of this law of force is reserved, in this connection, for the Celestial mechanics. 688. When in addition to the term, ivhich is inversely proportional to the square of the radius vector, the central force has a term inversely proportional to the cube of the radius vector, the path can be derived from the preceding section bij the principle of § 683. If the term of central force, which is inversely proportional to the square of the radius vector is the polar equation of the path is iZ^' _ « ^ ^ («2 _ 8 n^ B^pX) cos IB (if -a)-] = y/ (,,2 _ 3 X2^ B'^pt) Cos [B{if — a)sJ—l] == v/(8 12, B'-pt — d') Sin [B (y — a) sj — 1], when S2q is positive, therefore, the curve is finite ; it returns into itself if B^ is positive, but if B^ is negative it terminates at each extremity in an infinitely compressed coil about the origin. When S2q is negative, one portion at least of the path extends to an in- finite distance from the origin ; if, moreover, a is positive and & negative, but such that a'>Sn,B'^pt, another portion of the path is finite and terminates in the origin, — 388 — through an infinitely compressed coil, while the two infinite por- tions commence in such a coil ; if the negative B"^ is such that a^<%n,B''p\ or if a is negative as well as £2^, the curve only consists of the portion which extends from the coil to infinity. The time may be computed by the three formulae, which correspond to the three forms of (387i8), S[^-^(^-'2. + ^(^._Ji2.),.sin[i.(,-.)])] = —:;{87^l|P^ tan [*.?(,-«)], r„ [^-^ (^ ^4 + y/ (hI^. + i -^-'o) r Sin [i? (<;-«) y/- 1])] S[^>(^^2. + ^(j^_iii„),.Cos[i?(<;-«)v/-l])] = ^^_sc:i,B-^pt) ' CotliB{cp—a)^—l-]; the upper of the double forms of the first member applies to the case in which S2q is positive, and the lower to that in which S2q is negative. This case was partially developed by Clairaut. 689. The principle of § 683 may be extended to § 677, and among the resulting curves, that in which 7i is 2, deserves to be noticed from its simplicity, the equation of this case is 690. The laiv of central force, for tvJiich ilie integrals, involved in the equations of motion, can he expressed hj the elliptic forms u'ithout — 389 — amj special detenmudion of S2q ami p^, maf/ he reduced io two fjeneral forms of algebraia poli/nomicd besides other fractional forms. The first of these forms is in which m is either 2, 1, |, or |^. The second form is F^b, r "'-3 + ^3 r'"-3 + b. /--s + b, r-'"-3 + b r-2— 3^ in wliich m is either 1 or 2. In each of these cases the term which is inversely proportional to r must be omitted. 691. Iji the first case of the preceding section, when m is unity, the equation (07892) acquires the form „= r M_, . It is obvious from inspection that whenever a^ = b^, is positive, a portion of the curve extends to infinity; but when- ever «4 is negative, the curve is of finite extent. It is also apparent that whenever a^ — iB'-pl, is positive, a portion of the curve terminates in an infinitely com- pressed coil about the origin, that no portion of the curve can ap- proach the origin except through such a coil, and that when a is negative, the curve does not pass through the origin. If all the roots of the equation are imaginary, a^ and a must be positive, and the curve extends — 390 — continnoiisly from the origin to infinity. If the moduli of the roots are h and \^ and the arguments a and a-^, and if the following notation is adopted 2 a Og — {li? -j- 1x\) cii a^ J^- =: [p — lif -)- 4 p li sin^ k « , ^1 = {p — li^" -\-^p K siii^ h u^-, i>- = {fi — lif A^^.qll sin- h (i , B{z^[q — 1i^^ -\-iqh^ sin^ a^ «i, ^li (? — r) COSZ=:^^, and if ^q is the value of (3 when r vanishes, the equation of the curve is 2 pl{p — g)- cos' do ^^ ^ {p—q)cos-do ^^ " '^^ "^ ' T p tan (/q y/ ( 1 — sin^ / sin^ ^,) ^, (— cosec^^ i\ , i^ ) , Y^ (1 — sin- ^ sin- ^) — y/ (1 — sin^ i sin- ^y) ° V/ (sin^ ^ — sin-^ and the expression of the time is A A^ (f tan d^ (t — t) I p'^-\- ) ( t l w ^^1 '-^ = sm- ;' ^', ^ + (cos ;' — cos e) ly, (cot ;', ^) (cos y — cos g) cos 7 ^^^^[-1] / / cot^ 7 -f sin'' A ' y/ (cot'^ ;.' -]- sin- /) ' \ Vcot- /!/ -j- cos'-^i/ ' and the value of (/ — t) is derived from that of (9 — a') by multi- 7-2 plying by -r—., and interchanging / and e. The elliptic integrals disappear when the two real roots are equal. In this case, ^/^ is positive, and the curve is continuous from the origin to infinity. With the notation Pi-z^i^ -\- /r — Irh cos « = (r — li cos «)"-!- /r sin- a , E\=^]\-\- li- — 2 y'l li cos « = (;*! — /i cos u^" -\- li" sin- « , the equation of the curve is ri (qj — a) v'cf4 1 Tn,.[-i] ^^— ^'^Q^^^ 1 Tor.[-i] /^^ — ^ cos « (r -[- rQ + r ri and the expression of the time is given by the equation When r/4 vanishes, if rj is the real root of the equation (SSOgg), the curve consists of a single portion which extends from the origin to infinity when i\ is negative, in which case ^3 is positive. But if i\ and a^ are both positive, the portion extends to infinity, and 'i\ is its least distance from the origin ; if i\ is positive while ffg is negative, each extremity of the curve terminates in the origin, and Vy is its greatest distance from the origin. 50 — 394 — When cfg is jjositive, if tl and i are determined by the equations the equation of the curve is i?((p-a)v/a 3_ erf, A_^\±J^C^^\(ll^]^ ^^ B{r^ — B'^f , [_ij sin^ v/(rf -|-^^ — 2^ ViCOs20 + ^/^;r(,|qr^53:Ti?^7i cos 2 O] ^'^^ 2 i? sl{_r, (l — sin^ ^• siiV^ 6^)] ' and the expression for the time is (^_t) v/«3= (5 + ^)3^.^ — 2i?^\^ + 2i? tan |^v'(l—sin^-sin\4). When ^3 is negative, if (3 and i are determined by the equations tan^ I (3 = "^;^~ ? the equation of the curve is Oi+^) 27? _— J,6— 2^,^ J,[^ 4i?^r, '^J i? (r, + i?-^^ rp^ . r_ij sin ^ v^ in J^B' ^2B' r, cos 2 + ^[r.j(rf+^*+2^ViCos20] 2 ^ y/ E^i (1 — sin^ 2^" sin^ ^)] ' and the expression for the time is (^;_T)^_a3==(^_7?)?J^,;(3 + 2i?^,^— 2^tanidv/(l — sin2^sin2^). When all the roots of the equation (SSOgg) are real, if, beginning with the greatest, they are arranged in the order of algebraic magnitude, they may be denoted by 7\, r^, r^, and r^. If they are all negative, the curve consists of a single portion which extends from the origin to infinity. But if Vi is the only positive root, the curve consists of a single branch, which extends by the same law as that expressed in (39I20). If r^ and r^ are positive, while the other two roots are negative, the curve consists of one or two por- tions, according to the same principles which distinguish the forms of (39I25). If ^4 is the only negative root, and if a^ is positive, the curve consists of two portions, one of which extends to in- finity, and Tj is its least distance from the origin, while the other portion is finite and limited by the circumferences described about the origin as centre, with r2 and r^ as radii ; but if a^ is negative, one portion terminates, at each extremity, in the origin, and rs is its greatest radius vector, while the other portion is contained be- tween the limiting circumferences of which rj and r2 are the radii. If all the roots are positive and if a^ is also positive, the curve consists of three portions, one of which extends to infinity and ri is its least distance from the origin, a second portion is limited by the circumferences of which r2 and r^ are the radii, and the third portion passes through the origin at each extremity, and i\ is its greatest radius vector ; if a^ is negative, the curve consists of two portions, one of which is limited by the circumferences of which ?'i and ra are the radii, and the other by the circumferences of which rs and r^ are the radii. When «4 is positive, the following notation may be adopted. Ti — Tg = A tan i £ tan i i] , t'l — r2 = A cot i e tan J i]i, r^ — r^^ A tan h £ cot \ ^] , rg — 9\ = yl cot i c cot h i]i , -k Tt — £ ; which give tan?/ tun //i ' — 396 — or sin (ij, — t]) COS S = . , , - . sin (?/i + ,i) For the portion of the curve, which is contained between the cir- cumferences of which ra and r^ are the radii, the notation ra sin i 7^ = / sin i x, rg cos ^i]=ilco8ix, ^ ^ tan i A/ (n — ?•) ' gives r sin ^ (j^i -f x) -f sin 1 (j^i — x) sin /9 / sin i (7/1 + ,^)^ sin H'/i — ^/) sin^" The equation of the curve is, then, ^ (qp — «)v/a ,_ sinK>/.— ^/) g, ^, siny,s\n ^ (,j, — x) ^T sin^^fa — x) 1 2pllcosi sini(,/i_x) ' I sini(7/i— x)sini(j?i4-x) ^ sin^^^j^i+x)' J sin ^ (?/ — x ) y/ (sin y, cosec x) ^_ ^ ^ y' ( 1 — cos^" tan^ d) V/[sin //, sin x — sin^ i sin^ i (,^, -[- x) ] ''^^^ ^ [1— sin2^•^in2 J- (7/^+x) cosec //^ cosec x] ' and the expression for the time may be obtained from this value of {(p — a) by interchanging x and 1] and multiplying by -f^. The nature of the motion through the space exterior to the circumference of which 7\ is radius, and within the circumference of which r^ is radius, may be derived from equations (3965_i5) by changing r^ to r^ and r^ to 9-4^ and augmenting each of the angles t] and X by the magnitude tt. When «4 is negative, the following notation may be adopted, ri — rg =: A tan -| e tan i 1], H — ^4 = ^ t^n h £ cot g 1; , ri — i\=z A cot 2- € tan ^ i^i , ^2 — ^3 = -4 cot I £ cot 1 1^1, ^ == ^ TT — I c . The nature of the motion between the circumferences of which — 397 — r^ and ?-^ are the radii may, then, be expressed by the equations (3965_i5), provided that rg is changed to r^, and ?-2 to r^ and the sign of a^ is reversed. The character of the motion between the circumferences of which r^ and r^ are the radii, may be expressed by the same equations with the change of Vi to rg, and of rg to ?\, the reversal of the sign of a^, and the increase of each of the angles 1] and x by Ji. The elliptic integrals disappear when two of the roots are equal ; in this case, if Ti denotes one of the equal roots, and if 11^ is the quotient of the division of the first number of (08929) by (r — rif so that the form of E^ is B^ = /?2 r^ -{- hi r -\- h, the notation may be adopted Bl = /?2 ri -j- hi Vi -\- h, 2 h -\- hiT = 2 B sj— h t3in{& sJ—h) = — 2B ^ h Tan{6>^h), hi + 2 7^2 r = 2 i? s/—ho^ tan (L\iJ—h,) = — 2 ^ yZ/^a Tan (^2 >Jh), ^_in±^^±Ml= _ v/- 1 tan (Bi^^-1)= Tan (7?^ ^,) ; the equation of the curve is and the expression of the time is t — T=z6r^-\-ri^i. When «4 vanishes, if a^ is positive, the notation may be adopted ri — r^ = B^ tan^ j e , Ti — rsz= B^ cot^ i fc- , r^ cos^ it = I cos^ I X , rg sin^ h=:l sin^ ^ x , e z=: 2' TT — £ ; — 398 — and for the portion of the curve contained between the cu^cum- ferences of which rg and y-g are the radii, tan^ (}n — i^) = tan^ it ''-^. The equation of this portion of the curve is, then, (^ -^, ^ cj>se /^ _ COS^X g, ^^^, 2 pi I COS ^ COS X ' \ cos yJ ^ ^ ' I cosx — cose , [_ij // 1 -|- cos^ t tan^ ^ \ ■^ sin X y/ (cos" t — cos^ x) "^ y Vcos' i — siir ^ cot- x/ ' and the expression of the time is obtained from this value of (f/) — «) by interchanging e and x and multiplying by ^\ Upon the portion of the curve exterior to the circumference, of which Vx is radius, the notation r — ri = B" tan^ ( i tt — ^(1) = ^f^^ B\ gives for the equation of the curve {cp — a)sla, ^ gi.^ 2 B^ 2 pi cos i r^ — B' (ri — B-^){r^ + i?^) ^^[-c-s^:)^'] B + o in [- 1] ^ ^ V/ ('"i + ''i cos' / tan' Cjp) " sJlr,{\r^B^co%''i — {i\ — BY&m-'i)-] >J [^ir^ BHo^H — {r^—By s\n^ iV and for the expression of the time (^_^)sec^ = (ri4-52)3^,.<3 — ^'g,^ + ^|^V(l + cos=^^'tan2^) + 2 Z?2 Tant-iJ y/ ( 1 + cos^ ^ tan^ d ) . If ^^3 is negative, the notation may be adopted ri — rg = B'^ tan^ ^g , '>\ — ^3 = -^^ cot^ 2« ; — 399 — whrch, combined with that obtained from (39727_3i) by changing i's into ;-2 and rg into ri, gives (398;) for the equation of the por- tion of the curve contained between the circumferences of which Ti and ^2 are the radii, while the expression of the time is derived by the process of (398i3). But with the notation obtained from (398i6) by changing rj into rg and reversing the sign of 7>-, the equations (398i9) and (39825) become the equation of the curve and the expression of the time, upon the portion which is con- tained within the circumference of which r^ is the radius. The form of the central force which corresponds to the dis- cussion of this section is 692. If 7)1 is 2 in the first class of § G90, the expression of the central force is and the forms of the equation of the curve are obtained from those of §691 by changing r into r, and (cp — a) into 2{(p — a). But the expressions of the time require, moreover, the substitution for (390.6) of p — q ^ for (39I14) of t-r = J (^-^ ^ tan^-^^ 7^r?-^^^±"^-^n > for (392.3) of for (393y) of (/ — r)v^ — «,= ^eF.(3, — 400 — for (39823) of (? — TjVf/i — ^ian ^^^ , for (394i2) of for (39425) of for (396i8) with the form of (39G23) of {t — t) y/«i= i cosz 3^i<3, for (3975) of for (397.4) of t — T=^ -h 6, for (398n) and (39825) of {t — t) «3 = i cos i 9=, ^ , and for (3992_c) of {t — t) y/ — ^(^3 = 2 cos i^\6. 693. In the special case of § 692, in which F is reduced to its first term, so that two of the roots of (38929) are real and two are imaginary, so that the only portion of § 691, which is applicable to this case, is from (39I15) to (39823). In this case, moreover, one of the real roots is positive and the other is negative if b^^ is positive, so that the curve extends to infinity ; but if b^ is negative, both of the real roots must be positive, so that the circumferences which correspond to these roots are the limits of the curve, and S2q is negative and satisfies the condition -i2o>|-^f^(-2J,). — 401 — 694. In the special case of § 692, in -which F is reduced to its second term, so that the equation (88939) lias no imaginary roots of i^ when 4o^ b. >'^1p\ When h^. i^ positive, there is only one real root, so that the curve extends to infinity from the circumference, which is defined by this root. When h^, is negative, all the roots must be real, and the two roots, which are positive, define the circumferences which limit the extent of the curve. 695. If m is f in the first class of § 690, the expression of the central force is F=^ h^ r ^ -|- h.i r ^ -f- h^ i'~'^ -)- ^ —3 and the forms of the equation of the curve are obtained from those of §691 by changing /• into z-^, and 9) — u into §(9) — a). But the formula? for the time are more complicated, although they are still reducible to elliptic integrals. If, indeed, the expression for the time assumes the form ,_ _ C %z^ 696. In the special case of § 695, in which F is reduced to its first term, so that F^hr-^, the conditions of the form of the curve are the same with those 51 — 402 — expressed in (40023_3o), but instead of (400n,), the limitation of Hq when h^ is negative is 697. In the special case of § 695, in which F is reduced to its second term, so that the equation (08929) bas no imaginary roots of y' r^, when In the special case, in which F is reduced to its third term, so that F=h,r-i, the equation (08929) l^^^s no imaginary roots, when In each of these cases, when £1^ is negative, there is only one real positive root, so that the curve extends to infinity from the circumference which is defined by this root. When ii^ is positive all the roots must be real, and the two roots, which are positive, define the circumferences, which limit the extent of the curve. 698. If m is i in the first class of § 690, the exj)ression of the central force is and the forms of the equation of the curve are obtained from those of § 691 by changing r into y/;- and (^ — a) into 5 (9) — a). But if — 403 — the expression of the time assumes the form ^ — "^=[77- ^^a,~J-^a,z' + "i^ + ^) 699. In the special case of § 698, in which F is reduced to its first term, so tliat and in that, in which it is reduced to its third term, so that two of the roots of (3892o) are real for \J r, and two are imaginary, so that the only portion of § 691, which is applicable to this case, is from (39I15) to (39024). ^^^ this case, moreover, one of the real roots is positive and the other is negative if S2q is negative, so that the curve extends to infinity ; but if S2q is positive, both of the real roots must be positive, so that the circumferences, which correspond to these roots, are the limits of the curve, and in the former of these cases ^3 is negative and while in the latter case hi is negative and 700. In the second class of § 690, when m is unity, the equa- tion (37822) of the curve assumes the form SO that it can always be obtained from the expressions of (/ — r) in § 692, by multiplying either of those expressions by 4^>^ When, in this class, the curve terminates in the origin, it does not usually — 404 — pass through the condensed coil of § 691. The formula for the time is ;_^= f -= The form of the force, which corresponds to this case, is F=^h r-^+ h r-^ + h y-'+h r 5 701. In the special case of § 700, in which F is reduced to its third term, so that F=^' 4? one of the roots of (08929) is zero, and the condition that all the roots are real is i_^^ When /2q is negative, if h^ is positive, the curve extends to infinity, in the space exterior to the circumference of which the positive root of (08929) is the radius; hut if bx is negative, the curve extends from the origin to infinity, if two of the roots are imaginary, but if all the roots are real, one portion is exterior to the circumference of which the greater positive root is radius and extends to infinity, while the other portion is contained within the circumference of which the smaller positive root is the radius, and this portion passes through the origin. When Hq is positive, hi is negative, and the curve passes through the origin, and is con- tained within the circumference of which the positive root of (38929) is the radius. This case of force has been analyzed by Stader. 702. In the special case of § 700, in which F is reduced to its last term, so that — 405 — all the roots of (oSOgg) are imaginary when S2q and b are both positive. When /2,) is positive, therefore, b must be negative and the curve is contained within the circumference of which the pos- itive root of (oSQog) is the radius. When S2q is negative, if b is positive the curve extends to infinity in the space exterior to the circumference of which the positive root is radius; but if b is negative, the curve consists of two portions, one of which extends to infinity in the space exterior to the circumference of which the greater real root is radius, while the other portion passes through the origin and is contained within the circumference of which the smaller root is radius ; or it extends from the orio;in to infinitv. 703. When m is 2 in the second class of § 600, the form of the force is F=b,r-]-b.,?--^-\-b,i'-''-}-br-', and the equation of the curve can be obtained in each case from that of §092, hy niultipl^dng [t — t) by 2y/|, and changing t — t into (f> — a, and ;• into r^. If . = ;^ the formula for the time is ^~l\/\'a,z* + a,:^' V/ [^'4 *"* + 03-'' + a., 02 + «i - + «] 704. In the special case of 1-^ there are two imaginary roots of r when h^ G-tyr When wQq is negative, if b is positive the curve extends to — 406 — infinity in the space exterior to the circumference of which the real root of (oSOso) is the radius; but if h is negative, and if all the roots of (oSQgg) are also real and two of them positive, the curve consists of two portions, one of which extends to infinity in the space exterior to the circumference of which the greater posi- tive root is radius, while the other portion passes through the orio-in and is contained within the circumference of which the smaller positive root is radius ; but if neither of the roots is positive when ^ and il^ are both negative, the curve consists of a single portion w^hich extends from the origin to infinity. When il^^ is positive, h must be negative and the curve consists of a single j)ortion which passes through the origin and is contained within the circumference of which the positive root is radius. This law of force has been analyzed by Stader. 705. Another class of central force, in which the integration can be performed by elliptic integrals, corresponds to the form of the potential ^. __ h^ r'' '" 4- h. 1^ '" -\- b.j r- '" -\- b^ r'" -f- b in which m may be either 1 or 2. If, in these forms y ^,w ^ — / , Z''" = a^ z'^ -\- cIq z^ -\- 6(2 z" -^a^z^ a = {2.b^ 7-'"' + 2 ^3 r-^'" + 2 b. r '« + 2 ^^ r -f 2 h) -[2.n,^^ + ^p\){r-+hf, the equation of the curve assumes the form and the expression of the time is m — 407 — 706. The following graphic construction gives an easy geo- metrical process for tracing the various cases of limitation of the extent of the path described under the action of a central force, and especially for finding by inspection the effect of the values of S2q and j}^ upon the limits of the curve. If 1 v construct the curve of which the equation is which may be called iJie ^potential curve, draw the straight line of which the equation is and the points of intersection of the straight line with the potential curve give the values of x for the limits of the path of the body. The path corresponds to those portions of the potential curve which lie upon that side of the straight line, which is positive with respect to the direction of the axis of y. 707. A term of il may be omitted in the preceding construc- tion which is inversely proportional to the square of the radius vector, and its negative may be combined with that term of the equation of the straight line which determines its direction. The omitted term corresponds to a term of the force Avhich is inversely proportional to the cube of the radius vector, and which m.ay be represented by (08017) ; and the corresponding equation of the straight line is 708. It is evident from the preceding construction, that if the poteiiiial curve has no point of contrary flexure, and if its convexity is turned 408 — in the direction of the positive axis of y, the path of the lochj can only consist of a single portion tvhich may have either an outer or an inner limit, or it may have neither or both. This case includes all forces of the form F=hr- + l,, r in which h^ and m -\- 3 have the same sign. Bid if the potential curve lias no point of contrary flexure, and if- its convexity is turned in the direction of tJie negative axis of y, t/ic path of tlie body may consist of a single portion ivldcli has either an outer or an inner liiuit, or it may liave neitlier, or it may consist of two separate por- tions of ivldcli one lias only an outer and the otlier only an inner limit. This case includes all forces of the form (4O85), in which ^1 and niA^ 3 have different signs. 709. Those portions of the potential curve, in Avhich y and x simultaneously increase, correspond to the distances from the centre of action, at which the force is attractive, so that the convexity of the path of the body is turned away from the origin. The portions of the potential curve, in which y decreases with the increase of x, correspond to the distances from the centre of action, at which the force is repulsive, so that the convexity of the path of the body is turned towards the origin. Any point, therefore, at which the potential curve is parallel to the axis of x, and the ordinate is either a maximum or a minimum, corresponds to a distance from the origin, at which the central force changes from attraction to repul- sion, and the path of the body has a point of contrary flexure. 710. If for an infinitesimal value of r denoted by ?", il assumes the form 12 = /?: 2% the path of the body cannot pass through the origin if n-\-1 is positive or if k is negative, except in the former case, when p^ van- — 409 — ishes and n is positive while I2q is negative, or ?^ is negative while k is positive; but if k is positive and ii-\-2 negative, the external portion of the path passes through the origin, and after passing through the origin, the continuity of curvature is destroyed and the path becomes a straight line. 711. If for an infinite value of r, denoted by the reciprocal of/, II assumes the form (4O828), the path of the body cannot extend to infinity when n and Ic are both negative, or when n and S2q are both positive, or when n vanishes and but the external portion of the path extends to infinity when n is negative and k positive, or when « is positive and S2q negative, or when n vanishes and 712. If a line is drawn parallel to the axis of x at the dis- tance S2q from this axis, and assumed as a new axis of a^i, and if ^1 and ^2 ^i"® the corresponding ordinates, respectively, of the straight line (407i4) ^^d of the potential curve, the value of the angle, which the path of the body makes with the radius vector, is given by the equation V y. B which admits of simple geometrical construction. If z-^ denotes the subtangent of the potential curve upon the axis of x^, the projection of the radius of curvature of the path of the body upon its radius vector is which is constructed without difficulty. By the combination of these two constructions, the path of the body may be obtained with sufficient exactness for most purposes of general discussion. 52 — 410 — 713. When the origin is infinitely remote from the body, the forces o/" § 676 are jyarallcl, and the plane of motion is 'parallel to the direction of action, and the equation (0785) gives, if the axis of z is supposed to have the same direction with the force, — 2cot; A,% of which the integral is in which a is an arbitrary constant, which is always positive, and this is the equation of the path of the lody referred to the same coordi- nates with those of § 571. 714. In the case of a constant force, the preceding equation assumes the forms , . a a 2a ^ g sin ~ so that, in this case, the path is a parabola. 715. The velocity in the direction of the axis of x is 2^ sin ,^ := sin I V ( 2 ^2 — 2 12 ) == v^ ( 2 a ) , so that this velocity is constant, and The equation of the curve, expressed in rectangular coordi- nates, is — 411 — 716. If a potential curve is constructed by the equation (407io),in which y maybe changed into x, and il retained as a function of ^, the limits of the path of the body are defined by the intersection of the potential curve with a line drawn parallel to the axis of z at the distance (/2q -\- ci) from this axis. The por- tions of the potential curve which correspond to the path, lie in a positive direction from the intersecting line. 717. If the force of § 713 has the form the equation of the path is ^, ^ + ^ = v/ (^^ + 2 ^1 "'4 + 2 \ a) sin [{x,-x) y/ (- A.)] = v/(^^ + 2Z.,/2o+2J,«)Cos[g^-2-o)y/A] 718. If the force of § 713 has such a form that ^^^ — {z+hy the notation h = {k' + J^^){n, + a), gives, for the equation of the path, which is easily transformed into the forms, which are appropriate when the radicals become imaginary. 719. In the case of a surface of revolution^ and a force which is — 412 — directed to a point upon the axis of revolution, the notation of § 576 gives — = !i sin 1 = iL^ D, ", A=^uv sin ,^ = ir' Dt ", so that the elementary area described hj the projection of the radius vector upon the 2)lc('^ie of i^y is constant. 720. The notation of § 578 gives ^ V 2u^ (Si — Ho) — A^ ' ^ ^ M V 2 ?/ (.Q — .Qo) — ^2 > •and, in the case of parallel forces ■^-^ — -^-^\2u^{9. — Slo)—A^' jy u_ -^ Ag ^ ^ ~ w y/ [2 M=^ (.Q — ^4) — ^'] * 721. Upon the surface of revolution which is defined by the equation the path of the hody maJces a constant angle ivith the meridian curve. In the case of B^A, the path is perpendicular to the meridian, and is a circle of ivhich the plane is horizontal. Whatever is the value of B, for the point at which v vanishes u is infinite, while v is infinite when u vanishes. Ujmn any other surface of revolution about the same axis, the in- clination of the path of the body to the meridian arc is the same ivith — 413 — the corresponding inclination upon the surface of equation (4122i) at the common circle of intersection of these two surfaces. Hence the limits of the path upon the given surface of revolution are its intersections ivith the surface of equation iiv = A, and the path extends over that portion of the given surface^ ivhich is ex- terior to this surface hj ivhich the limits are defined. 722. In the case of a heavy body the equation (4122i) be- comes u^z = — . 723. In the case of a heavy hodg upon a vertical right cone, if the hodg moves upon the inverted p>art of the cone, the path has an upper and a loiver limit ; hut if it moves upon the part, v)hich is helow the vertex, the path has an uptper limit from which it extends doumivards to infinity. In this case, if the notation of (341i3) and (341i6) is adopted, if two of the roots of the equation r- (r — ro) = 25rsin^acoscc' are imaginary, which corresj)onds to ^r sin'' a cos « if h is the modulus and /:> the argument of one of the imaginary roots, and if 7\ is the real root, the notation r^ — hc':i-'-^=zB''c''''^-\ r^ — hc-i^^-^ = B''c-'-'^-\ r — rj = B^ tan^ h (p , — 414 — gives for the equation of the path upon the developed cone 2B ^(2rj^—2B'-cos2i) ^ {■lr^ — 2 B' co&2i) and the position of the body at any instant is defined by the equation -\-2B^ tan i 9 y/ ( 1 — sin^ i sin^ 9) . If all the roots of (4132o) are real, and denoted in the order of decreasing magnitude by ri, r.2, and r^, and if ri — r^ = B^tnn^tj, r, — r, = B'' cot'' [i, r — ri=:/':)Han^(T TT — |f/)), the equation of the path upon the lower portion of the devel- oped cone is 1" |-,.^ {r^J^By-co^H — 4.rlB'-] ^'^^ \' (^r,-\-By cos' i— 4. r^B'^ and the position of the body at any instant is defined by the equation {t — t) cos i y/ (2^ COS a) == (^ + ^) cos^z ^^,(p — 2B^,(p -\- \j^-\- B sm (p) ^ {1 -{- cos^ i tan^ (f). — 415 — The equation of the path of the body upon the upper portion of the cone is determined by the combination of the equations (414i3_i6) with ri — r — i?2 sni (p sni I r,-r^B-^ •2 ? sin^ aB\J {g cos a) :? +oTi[-i] / (1+ cos-^ i tan- y ) [0\-\-B-f cos- ^—4 /'i i?-] tanL V s/[r^(ry-\-B^ycosH—4:7iB^] V 4ri^2 j and the position of the body at any instant is defined by the equation {t — t) cos/y/(2^cosc«) =^ — Cj-\-B) Qo^iTf^w ior)$? 07^-' /I — sin i sin op A- 1 B e, o) — z B sm I cos cp \ / — ^ — : — ^-^-^ . ' ' ' y \ -\- sin I sin qp The path of the body upon the upper portion of the cone may be exj)ressed in a somewhat more simple form by the equations r = ?'2 sin^ (f -\- rg cos^ (p , and the corresjDonding formula for the position of the body at any instant is v/(2^cos«)(i'-T) = ^§^-2v'(ri-r3)f<9. In the special case, in which the roots i'2 and rg are equal, the path upon the upper portion is a horizontal circle, and the equation of the path upon the lower portion is J(d-(?„) = tan<-Y'(-i-;,)-\/3tan>-'>y'(-l-?,0, — 416 — while the position of the point at any instant is defined by the equation V/(2^cos«)(^-^) = 2v/(r + iro) + |v^(-ro)tant-Y'(--^-r}- 724. In the case of a heavy hod// upon the surface of a vertical paraholold of revolution, of ivhich the axis is directed doivmvards, the path has an upper limit, from ivhich it proceeds downwards to infinity. If (33625) is the equation of the paraboloid, and if z^ and — q are the roots of the equation. %pfjz{z — z^)=^-, the path of the body when is defined by the equations ^ — ^i = (%+^) tan^c;, and the position of the body at any instant is given by the equation I cos i{t — T ) d^TT = cos^ / 3=, (f — ^ j 9) -|- y/ ( cos^ ^ tan^ 9 -|- sin^ i sin^ ip ) . But when p>q, the path is defined by the equations ^ — 01— (,?i+;9)cot2 9), p — q — {2^-\-p)^\i\^i, — 417 — and the petition of the body at any instant is given by the equation In the especial case of the path of the body is the parabola, which is formed by the inter- section of the paraboloid with the vertical plane, of which the equation is ?/cos^=v/(4/ + ?/^), and the position of the body at any instant is defined by the equation {t — r)sj{p g)_^ u 725. Ill the case of a hcavij hod;j vpon the mrface of a vertical paraboloid, of which the axis is directed uinvard, the path has an upper and a lower lindt. If y; is negative, (ooGos) is the equation of this paraboloid, and if — z-^ and — x^ are the roots of the equation (416ii), they correspond to the limits of the path. The path of the body is defined by the formuLi) 5- = — ^'i cos^ (/) — .?2 sin^ i^) , (•?2 — p) «iii2 i=-Z2 — ^1 ? and the time is given by the equation 418 THE SrHERICAL PENDULUM. 726. When the surflice upon which the body moves is that of a sphere, the problem becomes that of the sphencal pendulum. In this case, the path has an upper and a lower limit. If the centre of the sphere is the origin, if it is the radius of the sphere, the limits of the path correspond to the roots of the equation If the roots of this equation are %, 2-2, and — p, and if the nota- tion is adopted z=.z^ cos^ (p -\- ^2 sin^ (f , (p + ^i) sin^/^-e'i — *2, the path is defined bj the formula «v/[2^a-+.,)] - ^ 5i> (1=1;,,,) + ^ * (|=iV,), and the time by the formula 727. From the equation (418io), it is easily inferred that ^1^2 + J^'' = P {^1 + ^2), that the sum of % and ^2 is always positive, and that p exceeds Ji. 728. It is apparent from the inspection of (41 821) that, if the mutual ratios of ^1 and the roots of (418io) are unchanged, the — 419 — time of oscilMion of the imididum is proportional to the square root of its length. 729. If the length of the pendulum and the sum of z^ and p are given, it is evident from (4I821) that the time of oscillation increases with the increase of i^ and is a minimum when i van- ishes, that is, when in which case the path of the pendulum is a horizontal circle. The time of oscillation in this case is rp '2nR Sj[2g{p^z,)j The mutual relation of ^; and Zi, which is here given by the equa- tion (4I826), is 2p^z-, + -, n whence This value is a minimum, when 0j ^ .3 = R, in which case which is, therefore, the greatest time of vibration ivhen the path of the j)endidum is a horizontal circle. It is easy to see that i cannot vanish for all values of the sum of ;:» and Zi, but that its least value is determined by the equation sm- 2 ? := 4 — - — -T—^,, ip + ^i) — 420 — whenever It is also evident that the least value of the sum of p and z\ which corresponds to any assumed value of ^ is given by (4]9ao), so that for awj value of i , ilie greatest time of vibration is ivMch increases tviih i, and is infinite lohen i becomes a rigid angle. When i is an octant, the value oi p -\- z^ in (4193o) is a maxi- mum, and the corresponding values o^ p -\- Zi and T are p -\-z^ = 2R 730. In the discussion of the form of the path of the pendu- lum, it is convenient to adopt the notation In the case of (4IO7), the equations of §726 and 727 give 9.. / ,,2 p2\ (^' ~l) " ^' — y - 27iR U, When s^i vanishes and T is the time of a complete revolution. Wlien U, = n, — 421 — and T is the time of a semi-revolution. The time of a complete revoluiion, ivhca the poidulum moves in a horkontal circle is so ihat it is proportional to the square root of the distance of the plane of revolution from the centre of the sphere. 731. When the path of the fendidiim deviates sligldtij from a horizontal circle^ so that i is very small, the notation ^1 + ^2 = 2^3=2 7^008(^3, gives «2 = (;' + 23)'-=— 3^— 2- *" A^ _ E- — zi ¥g—~Tz' ' —^ Z-C0S2 9, 732. AVhen the path of the pendulum deviates slightly from n great circle, so that the sum of z^ and z.^ is small, p is large and i is suKill, the formuli^ become, by neglecting the fourth and higher powers of / H.,-..)co.s2^ + ^=±iI^%-= ?/_ = 2 Jt — 422 — so that the vibration corresponds to a complete revolution of the pen- dulum. 733. When the pendulum passes very near the lower point of the sphere, so that Zi differs but little from 11, the neglect of this difference and its higher powers gives 2^:=. R COS 2 /, p:=z Pi -\- tan^ ii^R — z-^^ _^_4^^(^-^0sin^/, z^ R — 1R sin^ i sin^ cp — (7? — z^ cos^ ^^ , U^=.n + [cosec ^ 3^^ i „) _ if^' S, ( J n)] y'(2 - ^) ; so that the vibration corresponds to a little more than a semi-revolution of the peiidtdiim. 734. In the general case the vibration of the pendulum corresponds to an arc of revolution which exceeds a semi-revolution^ but is less than an entire revolution. When the velocity at the highest point is quite small, the case of § 733 occurs, but the arc of revolution, which cor- responds to a vibration, increases with the increase of velocity at the highest point. When the highest point is below the level of the centre of the sphere, the case of § 731 gives the highest limit of the velocity at this point ; but when the highest point is upon or above the level of the centre, the greatest velocity extends to infinity, which limit corresponds to the case of § 732. 735. The azimuth of the pendulum at any instant, is derived from the equation of § 726 in a form suitable for computation by means of the following formula? ; Zz:^ R COS ^^ , R (cos /9i cos /9._j -|- 1 ) p :=z R sec a cos di -\- COS 62 — 423 — -- = E"^ tan a ( cos i\ -\- cos i\,) = , - ,' - - , 9 . 1 -\- COS « COS ^., COS" I = — , 1 -j- cos a cos Oi sin -^ ^1 . • cos -A- /9., , 1 -f- COS ^1 COS 6., tan (/ — — - — ~ — ' , sin i7j COS a., . , , cos 6, -\- COS ^., tan //i =; cos a cos tl, tan // = — V /, ', ' sni /^i ' cos /./' = COS ^2 COS fli , coi = i n — i, ^r^-^ I _. V (cof^ q r — sin^ ?• cos^ (jp ) ^^ cos i cos fij sin d.j, ' . sin M tan do tan 4 0, tan 1 tan ?.2 = — , \ — —^ , tan c/j cos \i^ T tan ^. tan /.o =: — s — Tv-i s-^ — ^-^ 1 tan- ?/ ( 1 -|- cos- ^ tan-^ gi) ^ tan''^ i cos^ gi cos d.^ tan j(< tan ^ Q^ tan /, cos j[i cos // cos /^, -)- (1 — cos II cos ?/ cos ?/,) sin^ / sin- g; ' ^^ cos ^ sin ^ri tan ^] r ^A / cos- z cos'- jr^ tan"^ i9.> \ ~ 1 ^^ — tan 0, ~ L '' \ tan^ \ " ' ^/ ~ '^'^ ^' J -|- COS i COS jW tan (^2^*^-1- ^^i — ^^2 "h ^-3 — ^^4? and the arc of revolution for the complete vibration is -\- cos i cos ,a tan ^2 9^y ( i ^ ) • These formulae do not appear to differ from those of Guuer- MANN, although the reduction is more extended. They give with equal fjicility the area of spherical surface which is described by the arc of a great circle, wdiich joins the extremity of the pendulum to the lower point of the sphere. — 424 — MOTION OF A FREE POINT. 736. When a material point is unconstrained by any condi- tion, and is free to obey the action of any force whatever, its motion in any direction is simply defined by the equation 737. If the coordinates are assumed to be of the partial polar form in which z = the distance from the plane of .t\y , ^) = the distance from the axis of .?, (^ = the inclination of ^ to the axis of .t, the value of T (1G228) is, for the unit of mass. The corresponding values of oj (IGo^) are (IJ = * J 0)1=: 0)2 = ,/(f'; SO that 0)2 25 the double of the projection of the ladantaneous area, irhich is described bij the radius vector of the point, upon the plane of x y . The equation (I662) gives, then. n Tt is apparent from (oDu,) that ilie second mcmhcr of tJie lad equation . — 425 — is the moment, tidth reference to the axis of z, of all the forces lohich act upon the point. 738. If the forces are p-oportiomil to the distances from the centres from u'hich they emanate, the hody moves as if it ivere under the influence of a single force, acting hy the same laiu ivith an intensity equal to the sum of the intemities of the given forces, and emanating from a centre vMch is the centre of gravity of the given centres regarded as masses proportional to the intensities of their action. For, if the notation employed in § 128 is adopted, and if m denotes the sum of the intensities of action, the value of the potential is 12 J ra in which ^ is a constant and can be absorbed into the constant II with which J2 is connected in the equations of motion. It follows from § 685, that the path of the hody is, in this case, a conic section, of which the centre of gravity is the centre. 739. If all the forces are directed towards a fixed line, tlie area described by the projection of the radius vector upon a pUme perpendic- ular to the fixed line is proportional to the time of desciipiion. For the instantaneous area is in this case constant by the equation (42429) ? in which the fixed line may be assumed for the axis of ^, so that the second member shall vanish. 740. In the example of the preceding section, a peculiar sys- tem of coordinates may be advantageously adopted. This system consists of the sum of the distances from two fixed points of the given line, the difference of these distances, and the angle which is made by a plane passing through a fixed line, with a fixed plane which includes this line. If, then, 2/ = the sum of the distances of the body from the two fixed points, 54 — 426 — 2 5' ^= the difference of these distances, (f =^ the angle which the plane including the body and the fixed line makes with the fixed plane, 2 « = the distance of the fixed points from each other, 2 1/; = the angle which the two lines, which are drawn from the body to the fixed points, make wdth each other, k = the perpendicular drawn from the body to the fixed lines, 9 9 9 9 9 9 the values of /c,yj, and T are a ' tan 11^ = — , The corresponding differential equations derived from Le- grange's canonical forms (164i2) are i>,(^') = Z>,(Fy') = 0, qpiqi'' I ^qlpp'q' -\-qi3\q qv ^ ' ^ pl ' «" ' 91 The integral of (4262o) is in which B is arbitrary, and this equation expresses the proposition of §739, and gives , £ Ba^ ^ P p'fql' — 427 — A second integral of these equations, corresponding to the principle of living forces, is The sum of the equations obtained by multiplying (42622) by p\p', (426^) by {- •' ^^^ 9 7 9 5 piqi a^¥' SO that, in this case, the force is equivalent to a force tvhich emanates from an infinite axis of uniform extent, and is inversely proportional to the ciibe of the distance from the axis. 746. When the curve is given upon which a material point moves freely, the law of the fixed force is restricted within certain limits which it may be interesting to investigate. The geometrical conditions of the force are simply that it must be directed in the osculating plane of the curve, and the normal force must be equal to the centrifugal force of the body. By the adoption of the notation the equality of the force in the direction of the normal iV, or of the radius of curvature () to the centrifugal force is expressed by the equation 747. Since the preceding equation is linear, all the special values of £2^ hj tvhich it is satisfied, may he combined into a neiv value hy ad- dition or subtraction. Previously to this addition, each value of il^ may he multiplied hy a factor, which may represent the mass of the body, and if the factor is denoted by m, the value of m il^ will correspond to the whole force acting ujDon the mass, and it is, then, evident that, if M denotes the mass upon which the combined forces act, and V its velocity, the combined power is MV'=:Eiymv^), which expresses a condition identical mth the theorem of Bonnet. — 431 — 748. If a special value of 12^ is represented by /2o, and if £1^ satisfies the equation so that it is the potential of a force, to the level surfaces of ^vliich the given curve is a perpendicular trajectory, the complete value of i2i is ill ivliich f is an arhitrarjj function. It is apparent, then, that D.^ has an endless variety of possible forms in every special case. But each form corresponds to an arbitrary value of one of the constants of the given curve, or of some combination of those constants. 749. If the given curve is the parabola, of which the equa- tion is the values, which correspond to the arbitrary value of .Tq, are /22==log(.y — ,yo) + X (1.= while those, which correspond to the arbitrary value of ^u, arc f2o=2(.r-.r„)+|;; and it is interesting to observe that when, in this case, the arbitrary function of i22 is assumed to be constant, the value of the force is independent of x^^ and p as well as of ^o- The values, which correspond to the arbitrary value of ^j, are — 432 — 750. If the given curve is the conic section, of which the equation is ecos{(p — cpQ) = - — l, the values, which correspond to arbitrary values of (p^ are r ^ ■0 in which B^ = (^r' — {Fr — 9^f. When the arbitrary function of SI^ is assumed to be constant, the force is independent of e and P as well as of (pQ, and its law is identical with that of gravitation. 751. If the given curve is the cycloid determined by the equations ^— ^0 = ^(1 — cos^), X — .To ^^ Ji{^ — sin t5) ; the values which correspond to arbitrary values of a-Q are ^2 = ^' + ^ (^ + sin &), 1 ^2o = y—yo' in which ^ is to be regarded as the function of ?/, which is deter- mined by (432i8). 752. If the given curve is a circle of tvhich the centre is the ori- gin ivJiile the radius is arUtrarg, the potential of the force is an arUtrary homogeneous function of the reciprocals of x and y, tvhich is of the second degree. — 433 — This peculiar result is the more worthy of attention because it can be extended to the sphere, so that the iiotential of a force hy tvhich a body may move upon a sphere of a given centre hut of an ar- bitrary radius, is Ukeivise an arbitrary homogeneous function of the second degree of the reciprocals of the rectangular coordinates, of tvhich the centre of the sphere is the origin. These problems are fruitful of new subjects of interesting geo- metric speculation. CHAPTER XII. MOTIOX OF ROTATION. 753. If the coordinates of the points of a sj^stem are the partial polar coordinates of § 737, and if (p^ is supposed to refer to some point of the system, that is, to an axis connected with the system, from which the corresponding angles ^ are measured, so that the value of (f is 9 = To + ^. that of T becomes T=h^,[ln,{:4 + ^ + ^^^!^)]• Hence the equation (164i2) gives D,n = D,2,{m,q-',if\), the second member of which is the derivative of doul)le the sum of the products obtained by multiplying each element of mass by the 55 — 434 — area described by the projection of the radius vector upon the plane perpendicular to the axis of rotation. If this area is desig- nated as the rotation-area for the axis, it follows from (oOgs) that the derivative of the rotation-area for the axis is equal to the sum of the moments of the forces loitli reference to that axis. It is obvious that the mutual actions of the system may be neglected in obtaining the sum of the moments. If, then, all the external forces tvhich act xqion a system are directed towards an axis, the rotation-area for that axis tvill he described with a uni- form motion, which is the principle of the Conservation of Areas. 754. The rotation-area for an axis may be exhibited geomet- rically by a portion of the axis which is taken proportional to the area, and it is evident from the theory of projections that rotation- areas for different axes may be combined by the same laws with which forces applied to a point, and rotations are combined, so that there is a corresponding parallelopiped of rotation-areas. There is, then, for every system an axis of resultant rotation-area, tvith reference to which the rotation is a maximum, and the rotation-area for any other axis is the corresponding projection of the resultant rotation-area. The rotation-area vanishes, therefore, for an axis zvhich is perpendicular to the axis of resiUtant rotation-area. ROTATION OF A SOLID BODY. 755. In the rotation of a solid body, the axis of rotation does not usually coincide with that of resultant or maximum rotation- area ; and the relations of these two axes is of fundamental impor- tance in the investigation of the rotation. The determination of these relations depends directly upon the moment of inertia. The moment of inertia of a hody or system of bodies upon an axis is the snni — 435 — of the products obtained hj midtiplying each element of mass hy the square of Us distance from the axis. The distorting moment with reference to two rectangular axes is the sum of the products obtained by midtiplying each element of mass by the pivducts of its distances from the two corresponding coordinate planes. Let then 9n = the mass of the body, (^j^^=r=the distance of the element dm from the axis of^:*, which passes through the origin, - Ij'f=^ the reciprocal of the moment of inertia for the axis of p, ni J., = the distorting; motion of inertia for the two axes which form a rectangular system with the axis of p, which gives ^,. J in If, then, ^)^ is the angle which the axis of p makes with the direc- tion of q, the moment divided by the mass, becomes 5 = ^^X ('' '"'"^ ^^) ^ ^^ X (^' - "' '"'' ^^) ' — ^., (^^^^ — 2 j; cos (fy cos 9),) . If Ij, is set off upon the axis from the origin, its extremity lies upon a finite surface of the second degree, which is, therefore, an ellipsoid, and may be called the inverse ellipsoid of inertia. If the axes of this ellipsoid are assumed for the axes of coordinates, the values of J must vanish for each of these axes, that is, there is no — 436 — distorting inertia for these axes which mafj he called the principal axes of inertia. 756. When a body rotates about an axis, the rotation-area for an axis, which is perpendicular to that of rotation, is obviously proportional to the distorting inertia for these two axes. There is, therefore., no rotation-area for a principal axis of inertia proceeding from rotation about either of the other ttvo axes of inertia. 757. If ^p is the velocity of rotation about the axis of />, the corresponding velocity of rotation about the principal axis of x is (3;=:^;cosf/)^, and the corresponding rotation-area is m 6'j, cos (px the cosines of the angles, which the axis of resultant rotation-area makes with the principal axes, are then proportional to COSJp^ COSJP^ , COSJPj ■'■X -*!/ -^Z SO that this axis coincides with the perpendicular to the tangent plane of the ellipsoid which is drawn at the extremity of the axis of rotation. The plane of niaxinium rotation-area is, therefore, conjugate to the diameter of the ellipsoid tuhich is the axis of rotation, which theorem is given by Poinsot. 758. If the reciprocal of the perpendicular let fall from the origin upon the tangent plane of the ellipsoid is set off upon the perpendicular, its extremity lies upon a second ellipsoid, which may be called the ellipsoid of inertia, and of which the principal axes are the reciprocals of the principal axes of the ellipsoid of § 755, and are proportional to the square root of the principal moments of inertia. 759. It is apparent //iff?^ the tangent plane to the ellipsoid of inertia — 437 — ivhich is draivn at the extremiti/ of the axis of maxinmm rotation-area is perpendicular to the axis of rotation. It is also evident, that the axis of rotation is one of the principal axes of the section of the inverse ellipsoid of inertia, tvhich is made hy a plane passing thro%igh the axis of inertia and perpendicular to the common plane of the axis of rotation and of maximum rotation-area, tvhile the latter axis is one of the principal axes of the section of the ellipsoid of inertia, tvhich is made hj a passing plane through this axis perpendicular to this same common jjlane. 760. Although the fixed axes of coordinates may be assumed at any instant to coincide with the principal axes of inertia, the axes of inertia are nevertheless in constant motion from the fixed axes, and at the end of the instant dt, after coincidence, the axes of rota- tion, which coincided at the beo-innino; of the instant with the fixed axes of g and z, will not remain perpendicular to the fixed axis o^ X, but will deviate from perpendicularity by the respective angles ^'^dt and — ^'y dt. The rate of increase of the rotation-area for the fixed axis of x, which arises from the external forces is, therefore, m ^x n " y \u L-f which represents the ivell-known equations given hy Euler for the rota- tion of a solid body. If the rotation-area for the axis of jt? is denoted by m A^^,, the preceding equation may assume the form lD,a=.D,K~ A', a: (H- /;) . 761. If the equation (43722) is multiplied by 2 ^'^ and added — 438 — to the corresponding products for the other axes, the integral of the sum is m "^ I^^ which is simply tJie equation of living forces. If p is the semidiame- ter of the inverse ellipsoid of inertia, about which the solid is revolving at the instant, the preceding equation may be reduced to ROTATION OF A SOLID BODY WHICH IS SUBJECT TO NO EXTERNAL ACTION. 762. If a solid body is subject to no external force, the centre of gravity may be assumed for the origin. In this case the first member of (437.22) oi' (43729) vanishes, and the equation (438g) becomes 2) m or so that the velocity of rotation is proportional to the diameter of the inverse ellipsoid which is the axis of instantaneous rotation, which is given by POINSOT. 763. It follows from §§757 and 762, that, if q is the perpen- dicular let fall upon the tangent plane which is drawn to the inverse ellipsoid at the extremit}^ of the axis of rotation, q is the axis of maximum rotation-area, which is invariable when there is no external force, and that — 439 — ^', = &'p COS ^)p = hjl COS (fp ^hq=. -', SO that the velocity of rotation about the axis of maximum rotation-area^ as ivell as the distance of the tangent plane vjhich is draivn to the inverse ellipsoid of inertia at the extremity of the axis of rotation are invariable dming the motion of the solid^ which are propositions given by Poinsot. They might have been deduced ^yith faciHty from the geometrical theorem of § 759, without the aid of the equation of Hving forces, which might on the contrary have been derived, in the present case, as an inference from these theorems, and this was the elegant pro- cess of Poinsot. If the solid body has no translation, the inverse ellipsoid re- mains constantly tangent to the same plane which is that of max- imum rotation-area, and which touches the ellipsoid at the extremity of the axis of rotation. It is apparent, then, that in the motion of the solid, the ellipsoid rolls upon the fixed plane of maximum rotation-area, uithout awj sliding ; which is Poixsot's mode of conceiving this motion. 764. The instantaneous axis moves within the body in such a umg as to describe the surface of the cone of the second degree, of Avhich the equation is -•n{h-m=^- The base of this cone is an ellipse perpendicular to the greatest axis of the inverse ellipsoid ivhen q is larger than the middle axis, or impen- dicular to the least axis, when q is less thrrn the middle axis; and in either case the centre of the ellipse is upon the axis to which it is per- pendicular. When q is equal to either the greatest or the least axis, this axis becomes the permanent axis of rotation ; but when q is equal — 440 — to the middle axis, the cone is reduced to a plane which corresponds to one of the plane circular sections of the ellipsoid of inertia. The axis of maximum rotation-area describes within the body the cone of the second degree of which the equation is The common plane of the instantaneous axis of rotation and of the axis of maximum rotation-area is ohviously normal to the surface of the cone described in the hody hy the axis of maximum rotation-area, which defines the relative position of these two axes at each instant. 765. The position of the axis of maximum rotation-area is fixed in space, and, therefore, the path of the instantaneous axis of rotation in space is determined by the preceding property, and a distinct geometrical idea of the cone described by the instantaneous axis in space, is obtained by conceiving the cone described in the body by the axis of maximum rotation-area to be compressed into a line carrying with it the cone described by the instantaneous axis, in such a way as not to change the relative inclination of the two axes or the surface of the cone of the instantaneous axis. The algebraic definition of the cone of the instantaneous axis in space is obtained by assuming the axes of the inverse ellipsoid to be arranged in the order of magnitude as x, y, z, in which the cone of the axis of rotation has the axis of x as its central axis, and adopting the notation cos E^^=^ , Ml = ^ + ^-^? = ^^-^(l-f)(l-f): and similar equations for the other axes, in which it is unimportant that the angles E-, may be imaginary, but it should be observed — 441 — that i)[fy is the largest of the quantities, M^, 3Iy, and M^; the fol- lowing notation is also to be adopted. sin^ Ez= — sin^ E^ sin^ Ey sin^ E^ , ^=(i-f)(i-f)(i-|) = N/[(i-f)(i-f)(i-f)]. J.Uy ±J±y sin t] sm « = -. — -, , sin 1] ^ J/^ sin ;/ sin (5 = sin 2 sin ^ ; which give for each axis, if x,i/,z denote the extremity of the in- stantaneous axis upon the surface of the inverse ellipsoid, 7,2 ^ sin^ E= I^ (/ — M!) sin^ E^, D X p Dp ~^~ f — M'i' I^ Dx' sin^ E^ = f^^^^^Df ^ p^ — Mi If, then, If is the angle which the plane of p and q makes with a fixed plane passing through ^, the cone of the instantaneous axis in sj)ace is defined by the equation f sin^ y^ D v^2 -^ _Z_= />.^_|_ z)^2_|_ 2> ^2^ or ^' \\ — ^. — 445 — Tlie area about the axis of g is evidently described uniformly by the principles of § 753, so that if a = y y/ /, and if ^ is a constant, we have l'^ (^^ cos % -f" ^'y cos %) -\- a- 1 ^'^ cos f = (/^ — /|) I// + n i^ ^ =^ '^ ^^'' 4 • The equation (4883) gives, in the present case, (/2 _ ^1)2 ^/^^ ^2 ^^^^ ;,2 (^2 _ ^1) (^^^ _ ^^) ^ provided the constants /^ and 4 ♦'ii'© determined by the equations The elimination of i// from the equations (445;) and (445io) gives ^2 /;' =: 7,2 (/2 _ ^^ (/^ _ ^^) _ ^,2 ^^4 (^^ _ ^J2_ 773. The limiting values of 4 correspond to the vanishing values of /^, and, therefore, reduce the second member of (445i9) to zero. If these values are denoted by 4, 4? and — p, it is evident, from the form of the equation, that p is greater than I, while 4 and 4 are included between — /and -|-/. The equations for the sjDher- ical pendulum of §§ 726 and 727 may be directly applied to the gyroscope by changing z into 4 and *o? ^1? ^'^^ ^2 iiito 4> A? ^ii^ 4? which give, by (41824-27)? — 446 — With this notation and the equations derived from (418i3_i4)5 the expression of the time is and the equation of the path described by the axis of this body in sjDace is which admits of reduction by the process of § 735. 774. When the velocity n vanishes, the gyroscope is re- duced to a case of the spherical pendulum of which the length is 775. When the two roots /^ and 4 are equal, the path of the gyroscope k a horizontal circle. The values of 4? _ 2l(k — Q — ^"T" l-l, ' ^^^' — 2i(i-i,y ^, _ , / r(/ +4)(4- Q] op (^11^ A ^' — \ L 2l{l-l,) J -^'V 2/ '^Z- 779. When 4 is equal to — /, it is also equal to 4, that is, In this case 4 is algebraically less than — /, and the velocity of rotation which corresponds to this case is given by the equation which gives 2l(I + Q 780. When p is equal to — /, it is also equal to 4, that is, In this case 4 is algebraically greater than — /, and the veloc- ity of rotation which corresponds to this case is given by the equation which gives n^ _ ( I -I,) (I,-!,) _ (l-l ^ ( 1,-1,) /r l-{-li Z4-4 '■ 2I(I+h) ^2— -qi^^ i' — 449 — 781. If, in the preceding case, 4 is equal to the negative of /, it will also be equal to 4? that is, and, in this case, the elliptic integrals disappear from the equations, so that they become 4 = /,-(4 + /)Tan^[A(^-T)v/(/+/i)], and although the axis is constantly approaching the upper veriical, after passing the loiuer limit, it never reaches the upper limit ; and if it hegiiis at the upper limit it never recedes from it. 782. In the simplest form of the problem of tlie spinning of the top, the extremity of the hody is a point in the axis of revolution, tuhich is restricted to move, vMhoxit friction, in a horizontal plane. In this case, the equation (4449) is still applicable, as weU as (4467), provided that the moments of inertia are referred to the centre of gravity of the top, and that I denotes the distance from the centre of grav- ity to the point in the horizontal plane. The equation (4883) gives, in this case, with the notation of §772, {p-ll)v^' -^l\\^P n-I^ll)Vr^lr{p-n){k-h)\ and if w' is eliminated by means of (445;), The comparison of this equation with (445i9), shows that the limits of motion are the same as in the case of the gyroscope, and 57 — 450 — under the condition of the equality of l^ «ind 4? the extremity of the axis of the body describes a horizontal circle. The expressions of the time and of the azimuth of the axis are not, however, capable of expression by means of elliptic integrals, except in special cases, of which that of § 781 is one, and another corresponds to the case of ■'-X 783. When the horizontal plane, to which the extremity of the top is restricted, is not smooth, the problem is usually more complicated, although when the friction brings the lower extremity to the case of rest, it reassumes the form of the gyroscope, and this is the modification of the problem which has been investigated by PoissoN. In this case of the gyroscope, hoivever, the friction becomes an interesting feature of the proUem, and has a peculiar effect upon the limits to which the motion is subjected. Instead of the equation (4449), the rotation about the axis of the body decreases uniformly, which is ex- pressed by the equation The area described about the vertical axis is also described in this case, at a uniformly decreasing rate, which gives instead of (4467), {p — li)y\>' + {n — ^n)a-l^^n(^-{k — t^t). The power of the system is reduced by the friction about the body-axis, which is proportional to the angle 1, and by the friction about the vertical axis, which is proportional to \\^ . If, then, the mean values of i// and ^ for a small interval of time are denoted by T/V ^ncl ^„j, the equation of the preservation of power may be re- duced to {P - P,) v/^ + (^ if = If (P - li) (/^ + /„ + /„ , 451 in which 2i7«<=|J(V''cos?)-"^> n' , . .. n a^Vo , , = j^XjJ^t COS ^^ -JYJ2 Wmt- The combination of this equation with (45O23) gives It is obvious from this equation, that if the friction about the body-axis vanishes," the height, to w^hich the gyroscope ascends, diminishes at each oscillation. If, however, the friction about the vertical axis is destroyed, the height, to which the gyroscope ascends at each oscillation, increases when the body-axis is directed upwards in its mean position ; but this height diminishes when it corresponds to a position in which the centre of gravity is below the fixed extremity of the axis. In all intermediate positions, and when both the frictions remain, the increase or decrease of ascent depends upon the peculiar relations of the various constants. In the spinning of the top, the rounded point rolls upon the suj^porting plane, which induces an acceleration about the vertical axis which is the reverse of friction, and this is the principal cause of the ready rising of a top into the vertical position of apparent repose, known as the sleeping of the top. THK DEVIL ox TAVO STICKS AND THE CHILD's HOOP. 784. Contrasted with the motion of the gyroscope is that of a solid of revolution of which, instead of a fixed point of the axis, the circumference of a section drawn through the centre of — 45'2 — gravity, perpendicular to the axis is restricted to move upon a point. A convenient type of this class of motion may be found in the familiar toy called the devil on tivo sticks. If the friction is neglected in this case, and the notation adopted from the preceding problem of the gyroscope, the rotation about the body-axis is found to be constant, and the equation for the preservation of area about the vertical axis is, by a slight reduction, i^f' sin^ I -\-n cos ^ z= B, in which B is an arbitrary constant. The principle of power gives, by reduction, the equation 2 2 1/^' sin^ ^ -j- ^' =z II -\- « sin ^ , in which II is an arbitrary constant, and « is a constant which depends upon the form of the solid and the radius of the confined circumference. 785. The combination of (4529) and (452i4) gives sin^ ^ ^'' = {II -\- a sin $) sin^ ^ — (B — n cos If, from which it is obvious that, in the general case, sin ^ cannot vanish, that is, the hody-cixis cannot heeome vertical. 786. When B vanishes, and H is greater than a, we have the ordinary case of the devil on two sticks, and, in this case, there are three real values of sin l, for which the second member of ( 45220) vanishes. Two of these values of sin ^ are contained be- tween positive and negative unity, and one of them is positive, while the other is negative ; they give the limit of the motion of the axis, and correspond respectively to the cases in which the centre of gravity is below or above the point of dispersion, which latter is of course the actual case of the toy. In either case, the end — 453 — of the hody-axis describes a curve ivMch is similar in form io the figure 8, and the apparent ivant of rotation about the vertical axis arises from the repeated change in the direction of rotation which occurs at each successive return of the bodg-axis to the horizontal position. 787. When B vanishes, and H is greater than «, but satis- fies the inequality, H>^[knaJ — n'', the three values of sin^, for which the second member of (45220) vanishes, are all contained between positive and negative unity. The positive value is the upper limit of the inferior position of the centre of gravity, as in the preceding case, and as it would be if the inequality of this section were not satisfied, so that both the negative values were to become imaginary or equal. But the two negative values are the limits of motion, when the centre of grav- ity is higher than the point of suspension, and in this case tlic bodg- axis describes a ivaving curve, and continues to rotate in one and the same direction about the vertical axis, ivithout ever becoming horizontal, ivhich phenomenon usually occurs in the devil on tivo sticks, at the beginning of the game, and before it has attained a sufficiently rapid rotation to assume a horizontal position. When H satisfies the equation H=^{^na:f — if, the two negative limits of sin 'S, are equal, and correspond to a gyration of the body-axis about the vertical axis in a right cone. The motion which corresponds to a positive limit of sin I in this case can be expressed by means of elliptic integrals. 788. Whenever // satisfies the inequality — 454 — the body-axis may become horizontal with the centre of gravity above the point of suspension, and in this position its gyration is positive or negative in conformity with the sign of B. If, more- over, n is greater than B, the vibration of the body-axis from the horizontal position extends so fir as to reverse the direction of the gyration about the vertical axis ; but if n is less than B , the direc- tion of this gyration remains unchanged. When H satisfies the inequalitj^^ H, «i = 2„ (DJ, 2), f) = S, [X, (DJ, DJ,) Df^ f] , for by (19925) = ^, {DJ, i>„ //- DJ\ />, //) . The condition that ^ does not involve w gives algebraically, and the notation gives in combination with (4608) in which a may be substituted for its equal / in the second member. 802. The integrals of (4602_4) obtained with the omission of the forces of the inferior orders, admit of arbitrary variation of the arbitrary constants, so that if such variations taken with reference to arbitrary elements which may be denoted by y, and X, the cor- responding variations of (4602^) with the omission of the terms de- pendent upon W are — 461 — and similar equations for X which give SO that if Cl^'^ does not involve the time explicitly. 803. If X is the element of actual variation of the arbitrary constants when the inferior forces are introduced, which element may be expressed as the time when it is so connected with the arbi- trary constants, as not to cause ambiguity, the variations of the equations (4602_^), give D. t^ = D, '^, so that D^ ^ does not involve the time explicitly. When x and X are changed to «j and a,,, it is sufficient to retain i and k in the notation Cj,'^, so that it is apparent from (461i8) that By elimination from the equations represented in the preced- ing form, the value of Dt a^ can be obtained identical with that of (46O20), so that it is evident that B^^ does not contain the time ex- plicitly. It is also apparent that except when — 462 — in which case 804. The independence of B^l^ of the time in an explicit form, renders it possible to compute its value for any instant, and the value thus obtained is universally true. Thus in the especial case in which the arbitrary constants are the initial values of ?], w, etc., the values of i>i'^, computed for the initial instant, are easily seen to vanish when the k and i refer to different points of the system ; but when Jc and i refer to the 7]o ^i^cl ojq of the same point, the value of B^l^ is positive or negative unity, so that In the case of rectangular coordinates these equations become, for either axis, 805. The especial variation of the constant II may be de- rived from the equation (ITl;) which gives provided that t^ is intended to express the t which is involved in any of the quantities denoted by i]. This development of the variation of the arbitrary constants is taken from Lagrange. LAPLACE S METHOD OF PERTURBATION. 806. The values of to, ?;, etc., can be substituted from the first integrals directly in the first form of the second member of — 463 — (46O7), and the integral values of a^ which are then obtained can be introduced into w , -^^ , etc., as a second approximation to their values. Tliis mode of analysis is especially useful when the equa- tions of the first form are linear with reference to -Jj , to , and their derivatives. For in this case it is apparent that the functions de- noted by / are linear with reference to t], co, which may be demon- strated in the following manner. Let 1;,, w^, etc., be special values of 1]^ to, of which there must be as many independent values as there are equations expressed by (4602_4). The arbitrary constants «j- may then be such that the complete values of i] and w are (^ — ^i («i w,) ; whence the values of ojj assume, by elimination, the linear form in reference to i], co, "etc. The values of D^^fi, are then functions of t, and do not involve rj, a), etc. If i5^ ^ represent forces, which are also functions of the time, the integrals of (460;) can be com- pletely obtained. By the substitution of these values of «j thus obtained in the expressions of t], the complete values of i] are ob- tained, which often admit of useful modification, and the success of the method depends upon the skill with which this modification is efiected. 807. A special case of frequent occurrence in the problems of celestial mechanics is one in which The value of the integral in this case is, for a first approximation, tj =za cos at -\-a-i^ sin a i , — 464 whence o) :^ — a a sin at A^ aa^ cos a t, a = 9^ cos at sin a t =■/, czj = -J] sin « ^ -f- - cos at^=.fi. The values of the constants obtained by integration of (4G02_4), are increased to and (^i + -^ft{D^'^GO^at; so that the complete value of i^ is 9^ = 05 cos^^-j-ojisina/ ^ — / [D^ 'Fsin«/)-| / [D^^ Wcosat). 808. The disturbed motion of the ordinary projectile ex- hibits an easy example of change of form. In this case, by the introduction of rectangular coordinates in which the axis of a; is horizontal, and that of ?/ vertical, the equations are whence :c^at + a, + tf,D^W — f,{tD,W) =.at + a,+f?D,W, y = -hgi'+a,t-^a., + tf,D,W-f,{tD,W) — 465 — HANSEN S METHOD OF PERTURBATIONS. 809. If Vf denotes any function of the time and of the arbi- trary constants in the undisturbed orbit, its value in the disturbed orbit may be obtained, from the integration of the equation by the substitution of t for t after the integration is performed. In the performance of the integration, the arbitrary constants are to be regarded as variable, and the value of F^ in the undisturbed orbit is to be taken for the initial value of V^. This introdiicUon of T for t constitutes the first frinciple of H^vnsen's method of 'pcrtur- lotions. 810. The application of this method to the example of § 808, gives, for the values of x and y t x.=.at + a,-\-f^[{T — t}D^T], t 811. In the example of § 807, the value of 1] given by this method is i]=:a COS at -{- ai sin at / I sin a (t — t) D W^ I , in which the form of notation is slightly modified so that no subse- quent change of t to z5 is necessary. A case, which often occurs in connection with this example, is worthy of notice ; it is when D^ W =ih cos {m t -\-&), 59 — 466 — in which case the value of i] is 1] = a cos at -\- (Xi sin at -{- —^ .^ cos [m t-\-Ey In the special case of m = a, this value of rj becomes 7j z=. a cos at -\- a-^^ sin at -\- — t h sin (at -\- t). 812. If the function V increases with the time from negative to positive infinity, so that for all values of t there is an instant which may be denoted by ^, for which the un- disturbed value of V coincides with its disturbed value for the in- stant denoted by t. The corresponding value of s-^ is a function of both t and r, which may be introduced into V^ instead of t, but after this substitution all the changes in the value of V^ must arise from those of ^^, so that and the difterential equation for the determination of z^ is r ' T In the integration of this equation, t must be taken as the initial value of ;^^, whence, for the first approximation, T T — 467 — After the integration is performed, the value of is derived from that of z_ by changing % to t. 813. The disturbed value of any other function, U may be partially obtained by the substitution of ^ for ^, and, since the residual portion is obtained from the equation T T T by changing t for t after the integration is performed, and complet- ing the integration, so that U^ may be the value of U^^ when t vanishes. This introduction of the didurhed time, ivhich is denoted ly z, con- stitutes the second principle of Hansen's method of jjertiirhations, and upon the skilful use of the two principles thus developed, com- bined with an appropriate choice of coordinates, depends the suc- cess of this highly ingenious and original method. 814. It is obvious that, in the first approximation, so that the last term of (467n) disappears for this approximation. 815. If V is such a function that it can be expressed in terms of 1^, etc., without involving t.», etc., or t, it follows from §801, that the second member of (4658) vanishes, when t is changed to t, so that this must also be the case with the second member of the equation derived from (46637), D •^=^m^.4 — 468 — The value of the first member of this equation can therefore be obtained by the integration of the equation provided that the integration is completed in conformity with the previous condition. 816. If one of the arbitrary constants, which may be denoted by fi is so involved in V that in which K does not involve the time, or if the form of V is the corresponding term of the second member of (4684) is The corresponding term of the second member of (467ii), if U has the same form with V — ^ in (468ii), is 817. If one of the arbitrary constants, which may be de- noted by Y is so involved in U that U — 7 may be expressed as a function of V without explicitly involving y ov t, the corresponding term of (46 7n) is reduced to 818. The further development of the methods of perturba- tions depends upon the peculiarities of the problem to which they — 469 — are applied. But the example, to which they are most appropriate, is that from which they have derived their origin, the motions of the bodies of the solar system, so that their ampler discussion is re- served for the Celestial Mechanics. SMALL OSCILLATIONS. 819. When the motion of a system is restricted to small oscillations about a position of equilibrium, the quantities i], etc., may be supposed to be so small that the terms of T and i2, which are of more than two dimensions in reference to these quantities and their derivatives, may be neglected. The- value of T may, then, by (IGSg), be expressed in the form in which the quantities denoted by T^'\ are constant. If the values of i], etc., are supposed to vanish for the position of equilibrium, the derivative of S2 with reference to either of these variables vanishes for the same position, so that X2 must have the form in which the quantities, denoted by i2^'^, are constant. The equations of motion, derived from Lagrange's canonical forms, are, therefore, represented by that is, the?/ constitute a system of linear differential equations tvith constant coefficients. 820. It follows from the linear forms of these equations, that — 470 — the various systems of values by which they are satisfied, can be combined by addition into a new system. This is the mathematical expression of the important physical law of the possibility of 'the siipefyosition of small oscillations. 821. With the notation the equation (46928) assumes the form If, then, there are m of the quantities i] , etc., if — iv' is one of the values of D'^^ which satisfies the equation, expressed in the notation of determinants, any system of values of tJj is expressed by the equation 'r]i=zEiSm{nt-\-E,,), in which e,^ is an arbitrary constant, and the constants Ei have a common arbitrary factor. The mutual ratios of the quantities E^ are determined from the equations derived from (470io) by the substitution of — n^ for D'^, and E for i]. Hence, by §340, E^ is determined in the form E ^=E B^'^^^ , in which E,, is an arbitrary constant. 822. By the combination of all the values of n, the complete value of i].i is 7y, = ^„ [E, E^ %, sin {n t + £„)] ; but it is evident that only those values of n should be retained for which the values n^ given by (470i4) are real, positive, and — 471 — unequal. For all other values of ?z^, the time i would be intro- duced into the value of i/^ in such a way that it w^ould indefi- nitely increase. It is plain, therefore, that the only values of n, which can be retained in (47028), are those w^hich correspond to elements of stability, so that if the elements i] are selected with a due regard to the conditions of equilibrium, those which corre- spond to the unstable equilibrium will disappear of themselves with the rejection of the corresponding values of n. When the j^osUion of equilibrium is stable tvith reference to all of its elements, all the m values of n^ are real, ^positive, and unequal. 823. The forms of T and i2 of § 819, lead, by inspection, to the equations T]^ = Tl'\ and the equation (46928) gives, for each value of n, if n written as an accent indicates a special value of w, to which the functional form is ajDplicable. If ^„ is determined by the notation and if the equations, represented by (46928), ^^^ added together after being multiplied by F^"\ the sum is If, moreover, T^ denotes the value of T when rji is changed to IJ}"-\ the value of b«, given by integration, is '^,, = T„ sin {nt-\-t,,). The elements ^ thus obtained, correspond to the independent ele- — 472 — ments of stability which affect the position of equilibrium, and embody the true analysis of the various forms of oscillation of which the system is susceptible. When the different values of n have a common divisor, the oscillation is evidently periodic. This investigation of the theory of small oscillation coincides, in substance, with that of Lageange. 824. The importance and variety of the forms, in which oscillation and vibration are physically exhibited, give peculiar interest to the mechanical discussion of this subject. But the mode of analysis is so dependent upon the form of the phenomena, that the special researches are reserved for the chapters to which they are appropriate. A SYSTEM MOVING IN A RESISTING MEDIUM. 825. When a system moves in a resisting medium, the law of resistance may be regarded as dependent upon the velocity, so as to be the same for all the bodies, but it may vary by a constant factor from one body to the other. If this constant factor for the mass lUi is denoted by /",, and if V^ is the function of the velocity v.i, the resistance to the mass nii moving with the velocity v^ is /li F^. If, then, rectangular coordinates are adopted, the equations of motion assume the form The corresponding form of the equation for the determination of the Jacobian multiplier is, by §§ 402 and 451, A log ^J. = 2, [k, ^. n^^ M\ . — 473 — This equation becomes, when the motion is unrestricted in space, D, log ^fo ^ 2.^ [A", (2 ^ + A, V)\ ; when the motion is in a plane, A log ^lo = ^. [/", (| + A, f;)] ; when the motion is in a straight line, Al0g^fc=:^,(/^A,^)- 826. It is evident from the linear form of these equations, that the multiplier can he separated into factors, each of ivhich shall inde- pendently correspond to a term of Vj . 827. When the resistance is constant, and the motion in a straight line ; or when the resistance is inversely proportional to the velocity, and the motion is in a plane ; or when the resistance is inversely proportional to the square of the velocity, and the motion is unrestricted in space, the multiplier becomes unit?/. In either case of motion, a term of the corresponding form may be added to the resistance without affecting the multiplier. 828. When the resistance is proportional to the velocity, the value of the multiplier in the case of unrestricted motion is in the case of motion in a plane it is and in the case of the straight line it is ^^=Lc^^iK. GO — 474 — All these results, with regard to the multiplier, are derived from Jacobi. 829. When the resistance is proportional to the square of the velocity, the value of the multiplier for motion, which is unrestricted in space, is for motion in a plane, it is and for motion in a straight line, it is 830. The sum of the equations (47226), multiplied by nii x'i, is When Vi has the form the integral of this equation is * T—il= — :E,\_hm.M^a,t)-\. 831. When there are no external forces acting upon the sys- tem, the sum of the equations (47226) for each axis multiplied by W2j, if x^ refers to the centre of gravity, is ^,m,D\x,^-:E,i^n,hV:'-^. If the resistance is proportional to the velocity, the integral of this equation is ^i nii {D, Xg — A)^=z — :Si {nii ki x^) , — 475 — in which A is an arbitrary constant. If ki has the same value for all the bodies, the complete integral is kt in which B is an arbitrary constant. 832. The introduction of polar coordinates, and the substitu- tion of xl^'^ for the product of the area described by lUi about the axis of z, multiplied by the mass w?,, give for the corresponding equa- tions of motion I^,Af = DQ,nn—h^^D,Af. When there are no external forces, the sum of these equations is When the resistance is proportional to the velocity, the integral of this equation is D,^,Af=C—:E,{hA'^), in which C is an arbitrary constant, which vanishes if the area van- ishes with the time. If ki has the same value for all the bodies the next integral is ^,Af = B{l—e-'^). So that the rotatioiKtrea instead of being proportional to the time is pro- poHional to l — c-^\ hit the position of the axis of maximum rotation-area is not affected hy this uniform mode of resistance, which proposition is from Jacobi. 476 THE CONCLUSION. 833. In the beginning, the creating spirit embodied, in the material universe, those laws and forms of motion, which were best adapted to the instruction and development of the created intellect. The relations of the physical world to man as developed in space and time, as ordered in proximate simplicity and remote complica- tion, in the immediate supply of bodily wants by the mechanic arts, and the infinite promise of spiritual enjoyment by the contempla- tion and study of unlimited change and variety of phenomena, are admirably adapted to stimulate and encourage the action and growth of the mind. True philosophy begins with the actual, but may not remain there ; it yields sympathetically to the projectile force of nature, and earnestly forces its path into the possible, and even into speculations upon the impossible. But whenever the initial impetus is exhausted, the philosopher may not be content to remain stationary, or merely to turn upon his axis. He, then, descends to the world of sensible phenomena for new instruction and a stronger impulse. Let such be our method. In the present volume the attempt has been made to concentrate the more im- portant and abtruser speculations of analytic mechanics clothed in the most recent forms of analysis, and to make a few additions, which may not be rejected as unworthy of their position. Much, undoubtedly, remains imperfect and unfinished, for it cannot be otherwise in a science which is susceptible of infinite imj)rove- ment ; and much must soon become antiquated and obsolete as the science advances, and especially when we shall have received the full benefit of the remarkable machinery of Hamilton's Quater- nions. But it is time to return to nature, and learn from her actual solutions the recondite analysis of the more obscure problems of — 477 — celestial and physical mechanics. In these researches there is one lesson, which cannot escape the profound observer. Every portion of the material universe is pervaded by the same laws of mechanical action, which are incorporated into the very constitution of the hu- man mind. The solution of the problem of this universal presence of such a spiritual element is obvious and necessary. There is one God, and Science is the knowledge of Him. APPENDIX. NOTE A. ON THE FORCE OF MOVING BODIES. It is remarkable, that, notwithstanding the convincing argu- ments of Leibnitz, the force of moving bodies is ahnost universally introduced into systems of ancLlijtic mechanics as being proportional to the velocity, instead of to the square of the velocity. Some philos- ophers, in quite an unphilosophic spirit, have stigmatized the early discussions of this subject as a war of words, as if the eminent geometers who entered into it could have been so deficient in their powers of logic and analysis. The great objection to the propor- tionality of the force to the velocity is derived from the necessity which it involves of resiardins; force in one direction as beino; the negative of that which is in the opposite direction. On this ac- count, when a body or system rotates without any motion of transla- tion, its aggregate force vanishes, so that such a motion would seem capable of being produced without any expenditure of force, and this statement has actually been made in some works upon astrono- my. Leibnitz proposed as test propositions the transfer of motion from body to body in various foiTQS, in all of which he supposed the whole force to be transferred from one body to another of a dif- ferent weight without any external action. But it is evident from the law of preservation of momentum that such a transfer is im- possible, and, therefore, this test cannot be practically applied. If, however, in the case of the impact of an elastic body upon a — 480 — heavier one at rest, the striking body is held fast, as soon as it comes to instantaneous rest by the transfer of all its motion to the other body, the subsequent action of the elasticity must finally cause the body which is struck to move forward with a velocity inversely proportional to the square root of its mass. The external effort applied to the system in this case to hold the body at rest, arises from the force with which the elastic spring of the bodies is com- pressed, and is therefore an evidence of such a compression, and a proof that there has been an expenditure of force in its produc- tion, although the momentum of the system is not changed until the body is held. If, again, a splierical ball were to be impelled into a cylindrical tube of the same diameter, which terminates in an- other cylinder of a different diameter, but which containing a ball that exactly fits it, and if the included air acts as a compressed spring, it is easy to imagine such a mutual proportion of the parts and weights that the second ball shall leave the cylinder at the very instant when the first ball arrives at a state of rest, and when the air has returned to its initial density. In this case the whole living force of the first ball passes without increase or diminution into the second ball, and the momentum is not preserved. It is true that an external force is required to keep the cylinders in place, but this is a mere pressure, which is no more entitled to be regarded as an active force than is the centrifugal force, or any of the modifying forces which are represented anal3^tically by equa- tions of condition. Seeing, then, that by admitting the square of the velocit}^ to be the true measure of the force of a moving body, the fiction of negative force is wholly avoided, and the funda- mental principles of mechanical problems are reduced to their utmost simplicity, there seems to be sufficient reason to reverse the modern decisions, and return to the higher philosophy of Leibnitz. — 481 — . NOTE B. ON THE THEORY OF ORTHOGRAPHIC PROJECTIONS. For the convenience of students, the theory of orthographic projections is here condensed into a few simple formulse. The projection of a line a upon another line h is «j = a cos J. If many successive lines represented by f/,, are so united that each line begins where the previous line ended, and if the last line terminates where the first began, the sum of the projections is ^,(r^cosy=0. If there are four of these lines, and if the three first are mutually rectangular and parallel to the axes of x, y , and z, this equation becomes 2^ {a^ cos * ) + «4 cos ^^ = . But it is evident that a^ is the projection of — a^ upon the axis of X, whence a^ = — ffiCos^^, and if the subjacent 4 is now omitted as unnecessary, this equation gives cos^ = X^(cos^cos',), of which the equation l=z^,COS^«, is a particular case. These equations may be applied to the projections of plane areas, if each area is represented in a linear form by the length of a line which is drawn perpendicular to it. 61 ERRATA. Page For Read 12, axes axis. 154-1, ) and 1520 The signs of the second member; i should be reversed. 15^4 acute right. 268 I X,. SOu these those. 40i8 resultant resultant moment. 40^3 different lines opposite directions. 4I22 force resultant of the forces. 42^ 0' with 1 •eference to with reference to 0'. 51, X, x^. 5I4 y n- 55i4 POINT UPON A DISTANT MASS MASS UPON A DISTANT POINT. 57i3 4 f- 5722 i and 5728 1. 2 |. *59, cos-^ COS'j^. 5921 ; and 60,2 four eight. 5923 and 6O21 two four. 73,6 surface surfaces. 882 h 5.. 85c B^ D^- 8521 X 4 4. 86u 4:7t 4.71 K. 8622 l and 1 1 and 2. 8810 -^t) + ^D. 90i8 and 9O20 ff,a-l H^-\ 9O27 (cos (?« - -1) cos {m — 1) »/. 9I25 See note on page 356. 9824 89r 8922. 99io rn r". IOO7 independent of dependent upon. 101,2 twice + r + 2r. * This correction only applies to some copies. 484 ERRATA. Page 1^<],9, 16, 18 107^9 lllio llll3 llll3 llll7, 20 nii9 117,9 11722 117lB 11721 1207 12020 1211 1222, 1285 126u 1278 127.4 139,, U02« 140.,5 141i,;.2(, 148,, 14822 1493 1496 14?7 149j8 cos^^si 149,, 150.8 I5O30 twice 15O31 151i I5I3 I5I5 I5I5 15U For o O (104i9) to the form (107i6) ■ k 72 42,2 for another jioint of the body which is near the former point dele /ix=. iP similar to 119s 7t above put in order place tan sec" % cote takei^'^ -|- sin^ t] cos- 01 = cos^ a sec"^ 0) Read 6. Al,. (IO49) to the form (107i). k rmr'"--^ • 24. 4^22* arisinof from this motion. like. II83. about. but. in order that. plane. cot. BcB. V CB. sec *. tan £ . take i^'Z. COS^ 7/ . cot^ CO = cot^ a . ■ ij sin^ (jp cos'-^ d -\- sin"-^ // sin^ d cos'^ d — sin*^ »/ sin^ cp cos^ d -\- sin^ ?/ sin^ d cos- (jp. sin'^ 1] sin gj ""'"^ ■ 6'^ -\- tan /3 r — COS^ ^1 (Jp' cot qi' sin' jy sin' qp. + • d'—. — tan/3. d'—. — cos^/3, g)'. (cotg)' + ig)')- ERRATA. 48 Page iior JJea« — 2. 1759 ap^ 4>^ 178,3 B(rn) ^w'). 180, ®. ^.. I8O7 «: ^r. 190,, B g^. 191, /' /i. 191 The sections 366 to 369 should be limited vi<^n — 1. bj the condition that I9I24 a®;.... (%<%[ ^;^>. 191,5 (-)"«•:+. ( Jn + ^nm^l)6^" ^ 192^0 (_).+. ' (—)-». 192,3 m — 1 m — i. 192i5 (-)"+' / y„ „ + (i 4-1) (,„-!-„ 4. 1) 200i„ 199, 199a. 202,, 111 2033 340 839. 215^9 A, i>iCi. ^1^19,24, 22O3 Xi, X2 . . . . •^ 3^15 ^2 • • • • ^?l- 222i3 2OO2 215^2. 224i formal normal. 226n ;.— 1 I. 2272 (2IO31), the (2IO31). The. 2279 F i^. 228,, A ^^.. 228^1 i>.. 231^5 187k, 1893. 2333 216a 231,,. 2345,7 D:c, ^4)(^) i?., ^(ai)(»>. 238,9 ^A.. ^A. 239 »ja/(((D Jf. 486 ERRATA. Page 247^ 247^5 25005 258ij 261, 262,, 26220 2632, 2659 277.2^ 279; 2812X 28I3, 282^ 282i4_ 18. 28O20 287]o_ ig 287i2 289,9 3176 328-354 3289 33O4 34822 364j9 369-370 37730 39323 4262, 431i9 47231 Q uniform h P. P' sin (jp ^{9 — (^) Cot kt Ra correct 1 Sj 71 B rachy stochrone is confined 3299 589 siny 35935 Tachystotrope level sint-« g)^ .Q2 2>, Read ^. uniform and constant in direction. h. h_ V" ?r. 2a2. .+. cot. cosqp. y/[A(^_a)]. Tan. {kt — 'la). Ra CPo • o 9 k nearly correct. — a. fim. 3 K A + 2/.\ n. Brachistochrone. is not confined. 3293. 588. cos V . 3593. Tachistotrope. given. sint-iJ. ^'. ^,. ALPHABETICAL IXDEX. Abel, method of investigating the holo- chrone, ..... 356 Acceleration of rotating cylinder upon which a body moves, when it is uniform, ..... 254 of a point by a moving line, . 247 Action and Reaction, . . . .132 of moving bodies, . . . 1G2 principle of least, . . . .167 Andersox on rolling and sliding motion, 458 Archimedes, spiral of, described by ac- tion of central force, . . . 384 Area, constant area described by a point upon a surface of revolu- tion, 412 in the motion of a free point, . 424 constant, when all the forces are directed towards the axis, . . 425 of rotation, .... 433 conservations of, . . . . 434 of rotation for a principal axis, 436 of rotation when it Is a maximum for a solid, 437 of rotation expressed by Euler's equations for the motion of a solid, . . . . . .437 of rotation, its axis when it Is a maximum for the free motion of a solid, 439 of rotation described by the gyro- scope about the vertical axis, . 445 of rotation of gyroscojje affected by friction, . . . .450 of I'otation of the devil, . . 452 of rotation of the devil, when it vanishes, 452 Paoe Areas, principle of, in a moving system, . 458 Astronomical Journal, see Gould. Asymptote of the brachistochrone of in- finite branches, . . . . 333 Attraction of an infinite lamina, . 46 of an infinite cylinder, . 49 of a point upon a distant mass, 55 of a spherical shell, . 56 of a Chaslesian shell, . 58 of an ellipsoid, .... 69 of a spheroid, . . . . 88 Axis of rotation, 12 of rotation, instantaneous. 19 of gravity, .... 50 of principal expansion, 118 of Inertia principal, . 436 instantaneous in a body and in space, .... 439 B. Bailey on the force of resistance to the motion of the pendulum, . . 291 experiments on the motion of pen- dulum of various forms in air, . 311 Barnard on the gyroscope, . . 447 Bary trope discussed, . . . .370 Bernoulli, John, on the synchrone, . 373 Bernoulli, James's, lemnlscate, . . 380 Bertrand on the tautochrone, . . 364 Bessell on the resistance of the pendu- lum, 292 experiments upon the seconds' pendulum, ..... 298 BoBiLLiER catenary on cone, . . 153 catenary on sphere, . . .157 Bonds of union of a rigid system, . . 126 Bonnet, theorem of combination offerees, 430 488 ALPHABETICAL INDEX. Booth, elliptic integrals, . . .147 BoRDA, experiments on the seconds' ^len- dulum, 296 Brachistoclirone, . . . . .328 on the surface of revolution, . 334 Brass sphere vibrated by Bessell, . 298 spheres, cylinders, and bars vi- brated by Bailey, . . . 311 C. Canonical forms of the equations of mo- tion, 163 Catenary, 134 on surface of revolution, . . 143 on riiiht cone, .... 144 on ellipsoid, . . . . .154 on equilateral asymptotic hypcr- boloid, 159 curious relation to the motion on an hyperbola when the central force is proportional to the dis- tance, 385 CArCHY on elasticity, . . . .124 solution of partial differential equa- tions, 201 on differential equations, . . 214 Centre of gravity, . . . . .55 its position with regard to equilib- rium of rotation, . . .131 [resultant moment for, . . 133 motion of, . . . . .262 of systems, .... 458 of a system in a resisting medium, 474 Central force of gravitation, . . .43 in relation to tautochrone, . 323 in relation to brachistochrone, . 330 for a point moving upon a plane, 3 78 special cases of, which admit of in- tegration, 379 forms which admit of general in- tegration, . . . . .383 forms which admit of integration by elliptic integrals, . . .389 third form which can be solved by elliptic integrals, . . . 406 Centrifugal force, ..... 245 for brachistochrone, . . . 329 Centrifugal force of body moving on sur- face, . . . . . . Characteristic function of motion, its variation, .... Chasles's shell and its attraction, . and Gauss's theorem, trajectory canals, . . . . analogy of attraction and propa- gation of heat, . . . . definition of his thin shells, potential of his shells, . his ellipsoidal shell, . attraction of his ellipsoidal shell, . Circle rotating Avith a free moving body upon its circumference, upon which a heavy body moves, rotating in a vertical position, with heavy body moving along its cir- cumference, . . . . rotating with heavy body moving on its circiunference, . involute, with body moving along it against resistance, . involute, a case of tautochrone, a tautobaryd, . . . . described in a case of central force, horizontal, when it is in the path of a pendulum, .... great, when it is nearly the path of a pendulum, .... general law of description, section of ellipsoid of inertia de- scribed by axis of maximum ro- tation area of a solid, . path of the gyroscope, Clairaut on a case of the tachytrope, . motion of a body when the central force is inversely as the cube of the distance, .... Composition of forces, .... Conclusion, ...... Condition, equations of, . . . Cone, catenary upon, .... tautochrone of heavy body upon, brachistochrone of heavy body upon, ...... motion of heavy body upon. 377 162 166 58 61 63 64 65 61 70 76 251 255 259 264 274 325 372 379 419 421 432 442 446 365 388 40 476 24 144 322 341 413 ALPHABETICAL INDEX. 489 Conic section described when the central force is proportional to the dis- tance, described when the central force is inversely proportional to the square of the distance, described when many central forces act proportionally to the distance, . . - . general law of description, Conservation of power, . . . . of motion of centre of gravity, . of areas, ..... of power in motions of systems, Constants, variation of arbitrary. Continuity, solution of, in cases of resist- ance, . . . . . in the potential of nature, . Coordinates, peculiar case of, . Couple of rotations, .... of forces, . . . . . Cusps of brachistochrone. Cycloid the tautochrone of a heavy body, the base of a cylinder on which lies a tautochrone, meridian curve of a surface of revolution on which lies a tau- tochrone, . . . . . the brachistochrone of a heavy body, a tachytrope, .... conditions of description, Cylinder, attraction of, . containing a catenarj', rotating with a body moving upon a given line of its surface, . having a heavy body upon its sur- face, vibrated by Bessell, . vibrated by Baily, containing a tautochrone upon its surface, .... 385 38G 425 432 163 242 434 458 459 273 32 425 18 40 332 318 321 323 332 365 432 49 143 253 254 298 311 319 D. Derivative multiple, . Determinants, theory of, functional, . 196 172 183 62 Determinants applied to multiple deriva- tives and Integrals, . . .196 Devil on two sticks, . . . . 451 DuBUAT on the law of resistance of a medium, .... 292 experiments on the pendulimi against a resistance, . . 294 DtJPiN on orthogonal surfaces, . . .79 E. Economy dynamic, of nature, . . .168 Elasticity, . . . . . . 116 Electricity, statical, . . . . .44 Ellipse, spherical, . . . . 147 described by central force which is proportional to distance, . 385 described under the law of gravi- tation, 386 Ellipsoid, attraction of, . . . 69 Chaslesian shell, . . . .70 of revolution, attraction of, . 87 of closest approximation to at- traction of spheroid, . . .103 of expansion, . . . . 118 of reciprocal expansion, . .121 with catenary upon its surface, 154 with brachistochrone on its surface, 344 defining surface of the brachisto- chrone, . . . . .347 of Inverse inertia, . . . 435 of inertia, 436 Elliptic integrals for attraction of ellipsoids, 83 for the catenary upon the cone, . 147 referred to spherical ellipse, . 149 for the catenary upon the sphere, 157 for the simple pendulum, . .256 for tautochrone on a moving curve, 318 for tautochrone on a cycloidal cyl- inder, 321 for brachistochrone with parallel forces, 333 for brachistochrone on jiaraboloid, 337 for brachistochrone on inverted paraboloid, . . . .341 for brachistochrone on cone, • 343 for brachistochrone on sphere, . 346 for circular brachistochrone, . 354 490 ALPHABETICAL INDEX. Elliptic integrals for two forms of central force, 380 for third form of central force, . 406 for motion upon a cone, . .413 for motion upon a paraboloid, . 416 for motion upon an inverted para- boloid, . . . . . 417 for the time of spherical pendulum, 418 for the azimuth of the spherical pendulum, . . . .423 for forms of force directed towards axis, 428 for rotation of a free solid, . 442 for the gyroscope, . . . 446 for the top, .... 450 Epicycloid a tautochrone, . . .327 a brachistochrone, . . .331 path described under action of central force, . . . .379 Equation of tendency to motion, . . lis of motion, ditferential, . . • Sjg of equilibrium, . . . . Sjjo of orthogonal cosines, . . .1520 of instantaneous axis of rotation, 1 7j5 of rotation for cylinder, . . 23i4 of condition involved in those of motion and rest, . . . 26i2 of condition referred to normal, 27i5 of tendency to motion expressed by potential, .... 3431 of resultant, . . . . 372i of potential of gravity, . . 459 Laplace's, of potential, . . 463 Laplace's, modified by PoissoN, 492 of potential of an infinite cylinder, 4931 of relation of potential to its para- meter, 554 of Gauss, for action normal to surface, eOoi of attraction of ellipsoid in direc- tion of either axis, . . . 82o4 Legendre's, for attraction of ellipsoid, 83i2 of Legendre upon attraction, . 8610 of function for expression of the attraction of an ellipsoid, . . 8602 of attraction of a homogeneous ob- late ellipsoid of revolution, . 8730 Equation of attraction of a homogeneous j^rolate ellipsoid of revolution, . 88ig of function developed in cosines of multiple angles, . . . 89i3 of elementary functions of Legen- dre's functions, . . . 9324 of Legendre's functions in spe- cial form, . . . . .993 of theorem for development into Legendre's functions, . . IOI22 Laplace's upon Legendre's functions, 102,) Laplace's more general form of Legendre's functions, . . IO220 of potential of ellipsoid referred to centre of gravity, . . . lOSjo of Legendre's second function, IO49 of external j^otential of spheroid with the introduction of ellipsoid of nearest attraction, . .10724 for axes of nearest ellipsoid of at- traction, IO83 of potential for point near the spheroid, . . . . . IIO3 Laplace's, for spheroid which differs little from a sphere, . 1152? of ellipsoid of expansion, . . 1 1 83 of surface of distorted expansion, 119i3 of total expansion, . . . 12O27 of ellipsoid of reciprocal expan- sion, 12I11 of equilibrium of translation, . 12 730 of funicular, .... 138i4 of catenary, .... I3827 of extensible catenary, . . 14115 of catenaiy upon a surface, . 1420 of pressure of catenary upon a surface, 14 2,0 of catenary upon a surface of revolution, .... 14429 of arc of spherical ellipse, . . 14920 of total expenditure of action, . 1 6 2,1 of living forces, . . . . 163i4 Lagrange's canonical, of motion, 164i2 Hamilton's changes of La- grange's canonical forms, . 165o7 for characteristic and principal functions, . . . . 17I20 ALPHABETICAL INDEX. 491 Equation of determinants, . . . 173, linear solved by determinants, . 177 simultaneous differential, related to linear partial differential, . 199 differential in normal form, . 210^ partial differential for Jacobian multiplier, .... 215i2 oomuion differential for Jacobian multiplier, . . . . 216, of Jacobian multiplier for equa- tions of motion, . . . 237i(, of translation, . . . .2422 of time of describing a line, . 243.,; of centrifugal force, . . 245i8 of motion upon a rotating line, . 247oo of motion of a heavy body upon a moving line, . . . . 257^ of gain of power by motion of the line of support, . . . 25925 of motion of a fixed line through a resistance, . . . . 27I3 of motion against friction, . 273ir, of fixed force for tautochrone, . 317,, of tautochrone for central force, 323i5 of general brachistochrone, . 328i8 of brachistochrone for fixed force, 328^7 of brachistochrone for radius vec- tor and jierpendicular from origin in central force, . . . 33O15 of brachistochrone for parallel forces, 331,1 of brachistochrone on surface of revolution for central force, . 3242s of brachistochrone of given length, 34 7;jo of brachistochrone of given expen- diture of action, . . . 349i(, of the holochroue when the time is a given function of the poten- tial, 357i4 of tautochrone from Lagrange, 36I2 of tachytrope, .... 364i; of tachytrope for central foi'ce in resisting medium, . . . 3C64 of tachistotrope in resisting me- dium, 369,5 ofbarytrope, .... 370^1 of path of a point upon a surface with fixed forces, . . .377; Equation of path of a body when the force is central, .... 37802 of path of a body upon a surface of revolution with central force directed toward the axis, . 4129,17 of the spherical pendulum, . 41 817.01 of force for the description of a given curve, .... 43O20 of EuLER for rotation of a solid, 437oo of living force in a rolling solid, . 45 737 of sliding motion, . . . 4585 of variation of arbitrary constants, 46O30 of variation of initial values of va- riables .... 462ii^i2 of Hansen's method of perturba- tions, . . . 465^, 46625, 467i(, of small oscillations, . . .46927 of multijilier in a resisting medium, 4 72,1 of power in a resisting medium, . 474i5 of translation of a resisting me- dium, 474o7 of rotation in a resisting medium, 475ii Equilibrium, equations of, . . . 7 conditions of, . 29 stable or not, . . . . 30 of translation, .... 127 of rotation, 129 oscillation about position of. 471 Euler, integral, 91 note on erroneous notation. 356 on differentia] equations. 214 centrifugal force on the brachisto- chrone, 329 on the brachistochrone of central forces, 330 on epicycloidal brachistochrone. 331 error regarding the brachisto- chrone, 353 compound brachistochrone. 354 compound tautochrone. 358 tachytrope of heavy body. 364 tachytrope for parallel forces. 366 tachytrope of constant velocity in a given direction. 367 tachistotrope of heavy body, 369 tautobaryd of heavy body, . 373 path of body gravitating to two centres, 429 492 ALPHABETICAL INDEX. EuLER, equations of rotation of solid, . 437 rotation of solid, .... 443 Evolute of the parabola a tachytrope, . 368 Expansion, linear, . . . . .117 ellipsoid of, . . . .118 surface of distorted, . . .119 total, 120 Expenditure of action, . . . .162 F. Fontaine on tautochrone, . . . 362 Force, its origin, ..... 1 measure of, . . . . .2 of moving bodies, ... 4 of nature, 28 fixed, 28 expressed in form, . . .29 potential of, . . . . 29 temporarily fixed, . . .34 composition and resolution of, . 35 moment of, . . . . .38 couple of, . . . . . 40 in a plane, . . . . .42 parallel, 42 modifying, . . . . .124 internal, maybe neglected in trans- lation and rotation, . . .131 equal and parallel, in equilibrium, 132 principle of living, . . .163 of perturbation, . . . 459 of moving bodies, . . .479 central. See Central Force. centrifugal. See Centrifugal Force. Form, expressive of force, . . .29 French, weights and measures introduced, 293 Friction opposing n'iotion of a body, . .270 changing sliding to rolling motion, 458 Functional determinant, . . . .183 Funicular, . . . . . .134 G. Gamma function, . . . , .91 note on, 356 Gauss on action perpendicular to surface, 60 maxima and minima of potential of gravitation, . . . .62 determinants, . . . .173 Gould's Astronomical Journal, on partial multipliers, . . . . .231 on motion when force emanates from an axis, .... 428 Gravitation, jiotential of, . . . .43 potential for mass, ... 45 the type of equal and parallel forces, 132 its level surfaces, . . . 132 Gravity. See Centre of Gravity. GuDERMANN On Spherical pendulum, . 423 Gyration of the devil, .... 453 of the hoop, 456 Gyroscope, 443 H. Hamilton's characteristic function, . 162 on Lagrange's canonical forms, 164 modification of Lagrange's ca- nonical forms, . . . 165 principal function, . . .169 new method of dynamics, . . 171 quaternions, . . . .476 Hansen, method of perturbations, . 465 Helix, rotating with body moving upon it, 254 Holochrone, ...... 354 Hoop, motion of, 451 Hyperbola, determining the limits of mo- tion on a rotating circumference, 265 described by central force, . 380 described by repulsive central force proportional to distance, . 385 described by force of gravitation, 386 Hyperboloid equilateral asymtotic, con- taining catenary, . . . 159 defining limits of catenary upon other surfaces of revolution, . 160 homofocal with ellipsoid, . . 77 containing brachistochrone, . .347 Inertia of matter, 1 moment of, .... 434 Integral multiple, . . . . .197 of diiferential equations, . . 199 Integrals, systems of, ... . 203 elliptic. See Elliptic Integrals. ALPHABETICAL INDEX. 493 Integration of the differential equations of motion, 172 Involute of circle, described in a resisting medium, . . . . .274 a tautochrone, .... 325 Ia^ory on corresponding points, . . 70 Ivory sphere \ibrated by Bessel, . 298 Jacobi on Legendre's functions, . . 88 on determinants, . . .195 on normal forms of differential equations, . . . . .210 new multiplier, . . . . 214 principle of last multiplier, . . 228 on the motion of a body in a resist- ing medium, .... 376 on motion of a body gravitating to two fixed points, . . . 429 on motion of a system in a re- sisting medium, . . . .474 Jellett on the tangential radius of curva- ture of the brachistochrone on a surface, 347 on the brachistochrone of a heavy body in a resisting medium, . 353 K. Klixgstierxa's problem of the tachy- trope, 365 Lagrange, method of mechanical analysis, 9 canonical forms of equations of mo- tion, 165 on determinant of derivatives, . 194 on differential equations, . . 214 modification of Euleriax multi- plier, 232 on the tautochrone, . . . 359 familiar formula of the tautochrone, 361 on the rotation of a solid, . . 443 on the motion of a body gi-avitat- iug to two centres, . . . 429 on the method of perturbations by the variation of arbitrary con- stants, 459 Lagrange on small oscillations, . . 472 Lamina, attraction of infinite, . . .46 Lame', relation of potential to its parameter, 55 Laplace, equation for the potential of gravitation, . . . .46 equations modified by Poissox, 48 attraction of Newtonian shells, . 75 functions, 88 theorems on Legendre's func- tions, 102 equation for nearly spherical spheroid, 115 on the tautochrone, . . . 360 on the rotation of a solid, . . 443 method of perturbations, . . 462 Lead sphere, vibrated by Newton, . 293 Legendre, attraction of Newtonian shells, 75 attraction of ellipsoids, . . 83 theorems on the attraction of ellip- soids, 86 functions, . . . . .88 special form of functions, . . 99 Leibnitz on the force of moving bodies, 479 Lemniscate, described under law of cen- tral force, 380 Level surfaces, . . . . , 32 of gravity, 132 a syntachyd, . . . .375 Limits of brachistochrone, . . . 348 of body moving under central force, 40 7 of heaA y body on surface of revo- lution, 413 Linear equations solved by determinants, 177 partial differential equations, . 199 equations of small oscillations, . 469 Logarithmic spiral described by a body on a rotating straight line, . .251 described against resistance, . 274 a tautochrone, . . . .325 a tachy trope, . . . .365 described under the action of a central force, . . . .379 M. Maclaurin's attraction of ellipsolil, . 75 Mass defined, 2 Matter, inertia of, . . . . .1 494 ALPHABETICAL INDEX. Maupertius, action of a system, 162 principle of least action, 168 Maximum and minimum of potential, . 29 for equal and parallel forces, 132 of velocity of pendulum in a re- sisting medium. 283 Measures, French adopted, 293 Medium, resisting, .... 270 bracliistoclirone in, 350 holochrone in, , 359 tachytrope in, . . . 364 tachistotrope in, . . . 369 barytrope in, . . . 371 tautobaryd in, . 371 synchrone in, . . . 374 syutachyd in, . 375 systems moving in, 472 Method of multipliers, .... 25 Hamilton's, of dynamics, . 162 Lagrange's, of perturbation, . 459 Laplace's, of perturbations. 462 Hansen's, of perturbations. 465 Modifying forces, .... 124 Moment of force, 38 resultant, .... 39 of inertia, 435 Motion necessary to phenomena, 1 uniform, 2 measure of, . 2 tendency to, . 5 equation of, . 7 perpetual, impossible in nature, 31 of translation, . . 241 of a point, .... 242 of rotation, .... 433 of a system, .... 458 IMultiple derivatives and integrals, . 196 Multiplier, method of, .... 25 Jacobian, .... 214 principle of last. 228 for equations of motion, 236 for motion of a point. 244 for motion in a resisting medium. 472 N. Nature, forces of, Newton's shell. Newton's experiments on pendulum, . 293 path described when the central force is inversely as the cube of the distance, . . . .379 Normal form of differential equations, . 210 Notation of reference, .... 4 NuLTY on the hoop, .... 455 on rolling motion, . . .457 Nutation of rotation, .... 456 O. Orthogonal surfaces, . . . .79 Oscillations about position of equilibrium, 30 of a body on a fixed line, . . 246 of a bod}' on a uniformly rotating line, 248 on a rotating circumference, . 252 of the pendulum, .... 256 of a heavy body on a rotating cir- cumference, .... 266 of the pendulum when the resist- ance is proportional to the veloc- ity, 282 of the pendulum when the resist- ance is proportional to the square of the velocity, . . . -. 285 of the pendvdum with the medium, 287 of the pendulum when opposed by friction, 290 of the pendulum observed by Newton, 293 of the pendulum observed by DuBUAT, 295 of the pendulum observed by BORDA, 296 of the pendulum observed by Bessel, 298 of the pendulum observed by Bailey, 311 small, theory of, . . .469 Paper sphere vibrated by Dubuat, . . 295 Parabola, path of projectile, . . . 258 described while rotating, . .267 a tachytrope, . . . .368 described by law of gravitation, . 379 ALPHABETICAL INDEX. 495 Parabola, condition of description, 431 1 Paraboloid, brachistochrone on, 336 path of heavy body on. 416 j Parallel and equal forces, 132 Parallelopiped of translation. '' of rotation, 14 of forces, 36 of moments, 39 of rotation-area, 434 Parameter of potential, . . . . 54 Perpetual motion impossible in nature, 31 Pendulum, simple, 255 in a resisting medium. 281 seconds, of uncertain length. 313 spherical, 418 spherical, related to the gyroscope 446 Perturbations, methods of, . . . 459 Planetary perturbations, case of, . 463 465 Platinum sphere vibrated by Borda, 296 PoixsoT, analysis of rotation. 12 relations of axis of rotation and of maximum rotation-area, . 436 velocity of rotation about axis of maximum area, • . . . 439 on the rotation of a solid, . 443 Point, equilibrium of, . . . . 128 motion of, 245 Poissox, modification of Laplace's equa- tion, 48 theorem on Legexdre's functions 100 on the pendulum in a resisting medium, 286 on the top, .... 450 Pole of synchrone, 373 Potential, 29 of gravitation, . . . . 45 relation to its parameter, . 54 of spheroid, 99 of equal and parallel forces, 132 curve, 407 Power defined, 3 law of, 163 gained or lost by a moving line. 259 Pressure upon the brachistochrone, . 329 Principle of living forces, 163 of least action, . . . . 167 of last multiplier, 228 Progression, rotary, 456 Projectile, path of, ... . 410 disturbed, 464 Projections, theory of orthographic, . 481 PuisiEUX on the tautochrone, . . . 326 Q. Quaternions of Hamilton promise a new progress to analytic mechanics, . 476 R. Reference, notation of in tliis book, . 4 Residuals to express integral of central force, 380 Resisting medium. See Medium. Resultant defined, 36 vanishes in equilibrium of transla- tion, 128 Resultant-moment, 39 in relation to rotation, 130 of gravity for centre of gravity, . 133 RiCCATi on central force. 379 Rolling of solid, 457 Rotation, analysis of, . 12 combined with translation, . 16 instantaneous axis of, 19 tendency to, 40 of expansion, .... 120 equilibrium of, . 129 of line upon which a body moves about a vertical axis, . 261 motion of, 433 of a solid body, . . . . 434 Rotation-area, 433 in a resisting medium, . 475 s. Screw motion includes that of all solids, 19 Seconds pendulum, of uncertain length, . 313 Sections, conic. See Conic Sections. Shell, attraction of spherical, . . .56 attraction of Ciiaslesian, . 58 Chaslesian ellipsoidal, . . 70 Newtonian, .... 70 Sleep of the top, 451 Sliding motion, . . . . .457 Solid motion analyzed, . . . .18 rotation of, . . . .434 496 ALPHABETICAL INDEX. Solution of a partial differential equation, 199 of continuity in law of resistance, 273 Sphere, attraction of, . . . .57 having catenary upon its surface, 157 vibrated as a pendulum, . . 294 a synchrone, . . . .374 condition of descrijjtion, . .433 Spheroid, potential of, . . . . 99 which is almost an ellipsoid, . .110 almost a sphere, . . . Ill Spiral logarithmic path on a rotating line, 251 logarithmic described against fric- tion, . . . . . .274 logarithmic a tautochrone, . 325 a brachistochrone, . . . 331 logarithmic a tachytrope, . . 365 logarithmic described when cen- tral force is inversely proportion- al to the cube of distance . .379 path of the axis of a soUd, . 44 3 Stability of the funicular, . . . . 135 Stader, special eases of central force,. 379 central force inversely proportion- al to the cube of the distance, . 385 central force inversely proportion- al to the fourth power of the dis- tance, 404 central force inversely proportion- al to the seventh j^ower of the distance, 40G Straight line, attraction of infinite, . 52 rotating uniformly, with body mov- ing upon it, ... . 249 described by heavy body, . , 255 rotating uniformly about vertical axis, and described by heavy body, 2G2 rotating uniformly about an in- clined axis, and described by heavy body, . . . .269 a tachytrope, . . . .365 Superposition of small oscillations, . 470 Surfaces of the second degree homofoeal, 79 orthogonal, 79 of distorted expansion, . . 119 of revolution containing catenary, 143 Surfaces of revolution containing tauto- chrone, 322 of revolution containing brachis- tochrone, ..... 334 with point moving upon it, . 376 Synchrone, . . . . . .373 Syntachyd, . . . ' . . ,3 75 Systems of integrals, .... 203 motions of, ... . 458 motions in resisting medium, . .472 Tachistotrope, ..... Tachytrope, ..... Tautobaryd, ..... Tautochrone, .... compound, .... in Lagrange's form, restricted by Fontaine, Tension of the catenary. Time disturbed in Hansen's method. Top, spinning of, . Translation, analysis of, . combined with rotation, . tendency to, ... etjuilibrium of, . motion of, .... in a I'esisting medium, Trifolia of Stader, Trajectory of level surfaces, . . 369 364 . 370 316 . 358 359 . 362 139 . 465 449 7 16 . 37 127 . 241 474 . 379 32 Variation of the characteristic function, . 166 of a function of the elements of a determinant, . . . .180 rotary, 456 of arbitrary constants, . . 459 Velocity, 3 Vieille on the motion of a body along a rotating straiglit line, . . 262 Virtual velocities, principle of, . . .7 W. Weights, French, adopted, . . , 293 Wooden sphere vibrated by Newton, 293 UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. THE LIBRARY UNIVERSITY OF CALIFORNIA T/\a AMnirr va J^'o v.. isso. YOUNG FOLKS' LIBEAEY La Junta, Colorado. Established by one who belie\es that Property is a Trust; and who takes this way to return to the people a part of the gain that has come to him through the increase in value of land in La Junta. 3^-iemat/ca/ Sciences i-ibraiy D C ADXlLIARf T7?