t I \ THEORY OF SCREWS THE THEORY OF SCREWS A STUDY IN THE DYNAMICS OF A RIGID BODY. BY ROBERT STAWELL BALL, LL.D., F.R.S.", ANDREWS' PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF DUBLIN, AND ROYAL ASTRONOMER OF IRELAND. DUBLIN: HODGES, FOSTER, AND CO., GRAFTON-STREET. BOOKSELLERS TO THE UNIVERSITY. 1876. DUBLIN : f rmiefc Jtt tj-je SCwi&jeritg f BY PONSONBY AND MURPHY. TO MY FRIEND, FREDERICK J. SIDNEY, LL.D., I DEDICATE THIS BOOK, IN PLEASANT RECOLLECTION OF THE YEARS DURING WHICH I WAS A PROFESSOR AT THE ROYAL COLLEGE OF SCIENCE FOR IRELAND. CONTENTS. INTRODUCTION. PAGE. I. On the Canonical Form of a System of Forces applied to a Rigid Body, xvii II. On the Canonical Form of the Displacement of a Rigid Body, . xviii CHAPTER I. TWISTS AND WRENCHES. SECT. 1. Definition of the word Screw, . . . -. . . . . I 2. Definition of the word Twist, . " . . '." L .... 3 3. Instantaneous Screws, ..'.'..'. . . . 4 4. Definition of the word Wrench, . ' . ' ib. 5. Notation for Twists, Wrenches, and Twisting Motions, ... 5 6. Restriction of the Forces, ..'.'.'.'. . . . 7 7 The Energy of Position, . ' , ".. '.'.'. . . . ib. 8. Theorem, .'.'...'/ 8 9. Theorem, ib. 10. Theorem, ..'.". . . . . . . . 10 11. Concluding Remarks, n CHAPTER II. THE CYLINDROID. 12. Introduction, II 13. On the Virtual Co-efficient of a Pair of Screws, . . . . ib. 14. Symmetry of the Virtual Co-efficient, 13 15. Composition of Twists and Wrenches, ib. 16. The Cylindroid, 14 17. General Property, I 7 1 8. Particular Cases, 18 19. Form of the Cylindroid, . I 9 20. The Pitch Conic, ib - 21. Summary, 2O vill CONTENTS. CHAPTER III. RECIPROCAL SCREWS. SECT. PAGE. 22. Reciprocal Screws, -. . . .,'>*.,.., . . . 21 23. Particular Instances, . ' Y . "... f *' . . ib. 24. Screw Reciprocal to Cylindroid, . . .... . .22 25. Reciprocal Cone, . . . . . . . ' ib. 26. Locus of a Screw Reciprocal to Four Screws, 24 27. Screw Reciprocal to Five Screws, , ? , ' 25 28. Screw upon a Cylindroid Reciprocal to a Given Screw, . . .26 29. Properties of the Cylindroid, .' .-. , , . ' .- . . ib. CHAPTER IV. SCREW CO-ORDINATES. 30. Introduction, . . . . / . ' .' . . . . 28 31. Intensities of the Components, . . ,. . , . 29 32. The Intensity of the Resultant, . . -..' -3 33. Co-reciprocal Screws, . . ... . . . . . 31 34. Co-ordinates of a Wrench, .... . . . . 32 35. The Work done in a Twist, . . . .,. ,.,.... . . ib. 36. Screw Co-ordinates, . . . . . . . . 33 37. Identical Relation, . . . . . ..,.,...'. ib. 38. Calculation of Co-ordinates, .. . . \ . . . 34 39. The Virtual Coefficient, . . , . .'..'.. . 35 40. The Pitch, '.'..'.. . ib. 41. Screw Reciprocal to Five Screws, . . . . . . ib. 42. Co-ordinates of a Screw on a Cylindroid, . . ., 36 CHAPTER V. GENERAL CONSIDERATIONS ON THE EQUILIBRIUM OF A RIGID BODY. 43. The Screw Complex, 38 44. Constraints, 39 45. Screw Reciprocal to a Complex, . . . . . . ib. 46. The Reciprocal Screw Complex, 40 47. Equilibrium, 41 48. Reaction of Constraints, ib. 49. Parameters of a Screw Complex, 42 50. Applications of Co-ordinates, 43 CONTENTS. IX CHAPTER VI. THE PRINCIPAL SCREWS OF INERTIA. SECT. PACK. 51. Introduction, . . . . . . -45 52. Screws of Reference, , . .. . . , * . 46 53. Impulsive Screws and Instantaneous Screws, . ..... . 47 54. Conjugate Screws of Inertia, . . . . . . ..-.-.. 48 55. Determination of the Impulsive Screw, ...... 49 56. System of Conjugate Screws of Inertia, . . . . . . ib. 57. Principal Screws of Inertia, . . 50 58. Kinetic Energy, . . . 52 59. Expression for Kinetic Energy, . . . . . . 54 60. Twist Velocity acquired by an Impulse, . . . . ' . . 55 61. The Kinetic Energy acquired by an Impulse, . . >:. -56 62. Free Body, . . ... . ib. 63. Lemma, . . . . . ^ ,.,,,, f . . ._ . , 67 64. Euler's Theorem, .'.,." ib. 65. Co-ordinates of a Screw belonging to a Screw Complex, . . . 58 66. The Reduced Wrench, . * . ., . . 59 67. Co-ordinates of Impulsive and Instantaneous Screws, ... 60 CHAPTER VII. THE POTENTIAL ENERGY OF A DISPLACEMENT. 68. The Potential Energy of a Displacement, .' . . .63 69. The Wrench evoked by Displacement, . '. . . . . .64 70. Conjugate Screws of the Potential, .66 71. Principal Screws of the Potential, 68 72. Co-ordinates of the Wrench evoked by a Twist, .... 69 73. Form of the Potential, . '* 71 CHAPTER VIII. HARMONIC SCREWS. 74. Definition of an Harmonic Screw, . . . . . -73 75. Equations of Motion, 76 76. Discussion of the Results, 80 77. Remarks on Harmonic Screws, 82 X CONTENTS. CHAPTER IX. THE DYNAMICS OF A RIGID BODY WHICH HAS FREEDOM OF THE FIRST ORDER. SECT. PAGE. 78. Introduction, ..-..'.'. . . ' ". 8 3 79. Screw Complex of the First Order, . . -. . . ib. 80. The Reciprocal Screw Complex, ."..". . . 84 81. Equilibrium, . . " . '. . . . . 86 82. Particular Case, .'-.'-. J .' . . . 88 83. Impulsive Forces, . . . . * . . . ' . ib. 84. Small Oscillations, . ' 1 . ' . ' . ; . ; . . . . 89 85. Property of Harmonic Screws, .- ". . . ... . 9 CHAPTER X. THE DYNAMICS OF A RIGID BODY WHICH HAS FREEDOM OF THE SECOND ORDER. 86. The Screw Complex of the Second Order, . . . , . . 92 87. Applications of Screw Co-ordinates, . ... . . ib. 88. Relation between Two Cylindroids, . . . ... -93 89. Co-ordinates of Three Screws on a Cylindroid, . . ' . . 94 90. Screw Complex of the Fifth Order and Second Degree, . ,, r . 95 91. Polar Screws, 96 92. Properties of Screws and their Polars, 98 93. Pitch Complex, ib. 94. Screws on One Line, 101 95. Displacement of a Point, 102 96. Properties of the Pitch Conic, 103 97. Equilibrium of a Body with Freedom of the Second Order, . . ib. 98. Particular Cases, .105 99. The Impulsive Cylindroid and the Instantaneous Cylindroid, . ib. 100. Reaction of Constraints, 107 101. Principal Screws of Inertia, 108 102. The Ellipse of Inertia, . 109 103. The Ellipse of the Potential, ill 104. Harmonic Screws, 113 105. Exceptional Case, ib. 106. Reaction of Constraints, . . . , 114 CONTENTS. xi CHAPTER XI. THE DYNAMICS OF A RIGID BODY WHICH HAS FREEDOM OF THE THIRD ORDER. SECT. PAGE. 107. Introduction, . ....... .: ...;:>. . ' . . 116 108. Screw Complex of the Third Order, . ' < , ",' > . . 117 109. The Reciprocal Scre\v Complex, . . ... ,., . . .118 HO. Distribution of the Screws, . . '--.'' ib. in. The Pitch Quadric, ..... , . . .119 1 1 2. Screws through a Given Point, 122 113. Screws of the Complex parallel to a Plane, 125 114. Determination of the Cylindroid, . * . . . . .126 115. Miscellaneous Remarks, ... ... . . .128 1 1 6. Virtual Co-efficients, , . . ,.,- , v . f , , , .-, . . . . . 130 117. Four Screws of the Screw Complex, . . . . . ib. 118. Equilibrium of Four Forces applied to a Rigid Body, . . . 131 119. The Ellipsoid of Inertia, ./.,.,.., . . . 1. . 133 1 20. The Principal Screws of Inertia, ... . . . . . 134 121. Lemma, . 1 <'#' '. '-"'"'', .... 135 122. Relation between the Impulsive Screw and the Instantaneous Screw, ib. 123. Kinetic Energy acquired by an Impulse, " . . ... . . 136 124. Reaction of the Constraints, 138 125. Impulsive Screw is Indeterminate, . /. ib. 126. Ellipsoid of the Potential, . .. ,;! .-. ,* '* ... - ... . 139 127. The Principal Screws of the Potential, . . . . ' . .140 128. Wrench evoked by Displacement, . . . . . . . ib. 129. Harmonic Screws, . .141 130. Oscillations of a Rigid Body about a Fixed Point, . . .142 CHAPTER XII. THE DYNAMICS OF A RIGID BODY WHICH HAS FREEDOM OF THE FOURTH ORDER. 131. Screw Complex of the Fourth Order, 146 132. Screws Parallel to a Given Line, ib. 133. Screws in a Plane, ib. 134. Property of the Pitches of Six Co-reciprocals, . . . . 147 135. Another Proof, 149 136. Property of the Pitches of n Co-reciprocals, 149 137. Special Screw of the Complex, 150 138. Particular Case, ib. 139. Statics, ib. Xll CONTENTS. Chapter XII. continued. SECT. PACK. 140. Equilibrium of Five Forces, . . . : ' . . .'< . 151 141. Problem, . .152 142. Impulsive Screws and Instantaneous Screws, . . . . ib. 143. Principal Screws of Inertia, * *53 144. Application of Euler's Theorem, . * . . . > . 154 145. General Remarks, . . . . 155 146. The Screw Complex of the (n - i) th Order and Second Degree, . 156 147. Polar Screws, .^ .157 148. Kinetic Complex, . . . . < . . . . . 158 149. The Potential Complex, . . . . . . . . 160 150. Harmonic Screws, . . . . * ' ; . . . . ib. CHAPTER XIII. THE DYNAMICS OF A RIGID BODY WHICH HAS FREEDOM OF THE FIFTH ORDER. 151. Screw Reciprocal to Five Screws, . ' . . . . ^ . ... . 161 152. Definition of the Sexiant, . '. ' . '. '., -\ . . .. *' ./ . 163 153. Equilibrium, . ... . : . , . . =V . ./ ' . 164 154. Impulsive Screws and Instantaneous Screws, . .. . . . 166 155. Analytical Investigation, . , iir ' . r x ^7 156. Principal Screws of Inertia, .. ..^ .. -' -: . ' . . 168 CHAPTER XIV. THE DYNAMICS OF A RIGID BODY WHICH HAS FREEDOM OF THE SIXTH ORDER. 157. Introduction, . ; . i/o 158. Impulsive Screws, .....'.., . ib. 159. Theorem, 173 160. Theorem, . ib. 161. Principal Axis, ib. 162. Harmonic Screws, 174 INDEX TO DEFINITIONS. The Numerals refer to the Numbers of the Paragraphs. SKCT. \ Absolute Principal Screws of Inertia, . . . . . . .105 ! Amplitude of a Twist, . . ... / . . .. . . 5 Co-cylindroidal Screws, ..... . . , . . 99 ( Conjugate Screws of Inertia, ....... . 48 , Conjugate Screws of the Potential, ....... 66 Co-ordinates of a Screw, ......... 33 Co-ordinates of a Screw belonging to a given Screw Complex, . . 59 Co-ordinates of a Twist, ...... .. ... 32 'Co-ordinates of a Wrench, - . ...... , . ib. I Co-reciprocal Screws, -' . . . . . . . ',. . 31 ! Cylindroid, . . . . . . . . . 15 'Ellipse of Inertia, ...... ... . . no 'Ellipse of the Potential, '. . . . . ' . . . . 112 Ellipsoid of Inertia, . . . . . . . ... 135 Ellipsoid of the Potential, . . . . .140 Freedom of the n th Order, ......... 43 Harmonic Screws, . . ..... . . . 74 .Impulsive Cylindroid, .......... IO 6 Impulsive Screw, .......... 47 Impulsive Wrench, .......... 45 Instantaneous Cylindroid, ......... ,05 Instantaneous Screw, .......... 4 ^Intensity of a Wrench ........... 6 - inetic Screw Compl xiv INDEX TO DEFINITIONS. SECT. Pitch, . . l Pitch Conic, . . 19 Pitch Quadric, . "9 Polar Screws, . . . . * * Potential Screw Complex, . . . . . . ! 49 Principal Screws of Inertia, 45 Principal Screws of the Potential, 68 Reciprocal Plane, [ H* I * - I ; Reciprocal Screws, .-'.'".'. . ,- 2I Reciprocal Screw Complex, . . . . . 4 1 Reduced Wrench, v . - - , 6o Resultant Twist, . . ' . . . ..'-. - i 3 Resultant Wrench, 5 Screw, - . * Screw Complex, . . . . .' . 3 Screw Complex of the First Order, . . -.' . . ,. . 83 Screw Complex of the Second Order, . . . .... 9 2 Screw Complex of the Third Order, . . . . . * i Screw Complex of the Fourth Order, . . .' ' . . : ' . . H7 Screw Complex of the Fifth Order and First Degree, . ' . s , ' . . 95 Screw Complex of the Fifth Order and Second Degree, . '. ^ . 9 6 Screw Complex of the Sixth Order, . . . .'.,'. . '57 Screw Complex of the (n- i y h Order and Second Degree, . . . 158 Screw Reciprocal to a Cylindroid, . . .. . ." 2i Sexiant, . . . . ' . : X 5' Twist, . . . . ' Twisting Motion, . . - * '' Twist Velocity, ib - Virtual Co-efficient, ^ /' . . . . - : . ' . . . ^3 Wrench, 4 Wrench evoked by Displacement, . . . . 64 PREFACE THE Theory presented in the following pages was first sketched by the author in a Paper* communi- cated to the Royal Irish Academy on the i3th of November, 1871. This Paper was followed by others, f in which the subject was more fully 'de- veloped. The entire Theory has been re-written, and systematically arranged, in the present volume. We owe to the geometrical ability of Poinsot and Chasles the two fundamental theorems from which this subject takes its rise. To the labours of Pliicker, and his school, we are indebted for the theory of linear geometry, which receives a physical interpretation by the Theory of Screws. References are made in the foot notes, and more fully in the Appendix, to various authors whose writings are connected with the subject discussed in * Transactions of the Royal Irish Academy, Vol. xxv., pp. 157-217. f Philosophical Transactions of the Royal Society of Lon- don, Vol. clxiv., pp. 15-40. Transactions of the Royal Irish Aca- demy, Vol. xxv., pp. 295-327. XVI PREFACE. this book. I must, however, mention specially the name of my friend Professor Felix Klein, of Munich, whose private letters have afforded me much valua- ble information, in addition to that derived from his instructive memoirs in the pages of the Mathema- tische Annalen. To my friends, Rev. ProfessorTownsend, F. R. S., Professor Everett, Dr. Tarleton, F. T. C. D., Pro- fessor Niven, and Professor Casey, F. R. S., my thanks are due for many useful suggestions and a little friendly criticism, while to the two first-named I am further indebted for their trouble in looking over the proof-sheets in their passage through the press. My grateful thanks are also due to the Board of Trinity College for a liberal contribution towards the expenses of publication ; and to the Council of the Royal Irish Academy for kind permission to use the wood engravings which illustrated the original Papers published in their Transactions. ROBERT S. BALL. THE OBSERVATORY, DUNSINK, Co. DUBLIN, 3 1 st December, 1875. INTRODUCTION. THE Theory of Screws is founded upon two well-known theorems. One relates to a system of forces acting on a rigid body; while the other relates to the displace- ment of a rigid body. Although these two theorems are to be found in many treatises on mechanics, yet a discussion of them here, so far as they are necessary for our purpose, may be useful. I. ON THE REDUCTION OF A SYSTEM OF FORCES APPLIED TO A RIGID BODY TO ITS CANONICAL FORM. The Canonical Form. It has been discovered by Poin- sot* that any system of forces which act upon a rigid body can be replaced by a single force, 'and a couple in a plane perpendicular to the force. Thus a force, and a couple in a plane perpendicular to the force, constitute what may be called the canonical form of a system of forces applied to a rigid body. It is easily seen that all the forces acting upon a rigid body may, by transference to an arbitrary origin, be com- pounded into a force acting at the origin, and a couple. Wherever the origin be taken, the magnitude and direc- See Appendix I. b INTRODUCTION. tion of the force are both manifestly invariable ; but this is not the case either with the moment of the couple or the aspect of its plane. The origin, however, can be always so selected that the plane of the couple shall be perpendicular to the direction of the force. For at any origin the couple can be resolved into two couples, one in a plane containing the force, and the other in the plane perpendicular to the force. The first component can be compounded with the force, the effect being merely to transfer the force to a parallel position; thus the entire system is reduced to a force, and a couple in a plane perpendicular to it. The Canonical Form is Unique. It is very important to observe that there is only one straight line which possesses the property that a force along this line, and a couple in a plane perpendicular to the line, is equivalent to the given system of forces. Suppose two lines possessed the pro- perty, then if the force and couple belonging to one were reversed, they must destroy the force and couple belong- ing to the other. But the two straight lines must be parallel, since each must be parallel to the resultant of all the forces supposed to act at a point, and the forces act- ing along these must be equal and opposite. The two forces would therefore form a couple in a plane per- pendicular to that of the couple which is found by com- pounding the two original couples. We should then have two couples in perpendicular planes destroying each other, which is manifestly impossible. We thus see that any system of forces applied to a rigid body can be made to assume an extremely sim- ple form, in which no arbitrary element is involved. INTRODUCTION. XIX II.- ON THE REDUCTION OF THE DISPLACEMENT OF A RIGID BODY TO ITS CANONICAL FORM. Problem. Two positions of a rigid body being given, there are an infinite variety of movements by which the body can be transferred from one of these positions to the other. It has been discovered by Chasles* that among these movements there is one of unparalleled simplicity. The demonstration of this theorem is the object of the present section. The Composition of Rotations about Intersecting Axes. Suppose a body receive a small rotation through an angle a about a certain axis, and another small rota- tion through an angle j3 around a second axis inter- secting the former one; then the position ultimately attained could have been reached by a single rotation from the initial position about an axis appropriately chosen. Let OA and OB (Fig. i) represent the directions of the given axes, while their lengths are proportional to the angles a and |3, the directions of the rotations being such that if an ordinary screw were placed with its head at O, and its axis along OA, then the direc- tion of the rotation which would make the screw advance from is the direc- tion of the rotation See Appendix I. XX INTRODUCTION. indicated by OA, with a similar convention for OB. Completing the parallelogram OACB, we shall prove that a rotation around OC y through an angle propor- tional to the length OC, will have precisely the same effect as the two given rotations. Consider any point P of the body which lies in the plane of the axes. The rotation OA will depress the point P below the plane of the paper along the normal to a small distance which is propor- tional to the product of OA, and the perpendicular PQ ; that is, proportional to the area of the triangle POA. In the same way the rotation around OB will raise P above the plane of the paper to a distance which is proportional to the area viPOB. The joint effect will be to raise P to a distance above the paper propor- tional to the difference between the areas of the triangle POB and POA. that is, to the area of POC; but this is precisely the same effect as would be produced by a rota- tion around OC through an angle proportional to OC. To prove that POC = POB - POA : draw PR parallel to OA ; then OAR = GAP, and BRC^BPC, whence POA + PBC = OBC ; also we have POA + POB + PBC = POA + POC + OBC, since each side represents the area of the figure OAPBC ; therefore POB = POA + POC. or POC = POB -POA. The rotation around OC must, therefore, produce precisely the same effect on every point in the plane as is produced by the joint effect of the rotations around OA and OB. INTRODUCTION. xxi and hence it follows, that the two rotations about OA and OB can be replaced by the single one about OC. The correspondence between the solution of this prob- lem and the principle embodied in the parallelogram of force should be noticed. We see that rotations about intersecting axes are compounded by the same rules as in- tersecting forces. Composition of Rotations about Parallel Axes. We shall now consider the case in which the two axes A and B, about which the body receives small rotations a and j3, are parallel. Divide the perpendicular distance ^between the parallel axes A and B in the inverse proportion of a and /3, and draw a line C parallel to A and B through the point thus obtained. We shall show that a rotation around C through an angle a + ]3 will be precisely equivalent to the two given rotations. For consider any point P in the plane at a perpendicular distance x from C. Then the distances of P from A and from B are respectively x + d-rt and x - d-^^ a + 3 a + ]3' The effect of the rotations about A and B will, therefore, be to raise P above the plane of A and B to an amount but rotation about C through an angle a + )3 would have had precisely the same effect, and the same will be true for every other point in the plane besides P. We thus see that rotations about parallel axes are com- pounded by exactly the same laws as parallel forces. Translations* The rule for the composition of parallel rotations would not apply if the two r otations were xxii INTRODUCTION. equal and opposite. We proceed to consider this case. Let the angle of rotation be a, the axes A and B, and their distance d. Let x be the distance of any point P from A ; then the rotation about A elevates P above the plane of A and B to a distance ax. The rotation around B depresses P below the plane of A and B to a distance a (x + d). The net result, therefore, is that P is depressed below the plane of A and B to a distance a - Pa cos *m +pp sin 2 ;;/. z 2 = (p a ~ Pi) si n m cos m - A = / - m, h = Zi - z 2y which the axis makes an angle w with the axis of x is k = #> cos ! at -f sin* w. The writer was informed by Dr. Felix Klein that Pliicker had also constructed a model of this surface. Pliicker does not appear to have contemplated the mechanical and kinema- tical properties of the cylindroid, with which alone we are concerned ; but it is worthy of remark that the distribution of pitch which is presented by physi- cal considerations is exactly the same as the distribution of parameter upon the generators of the surface, which was fully discussed by Pliicker in connexion with his theory of the linear complex. The name cylindroid was suggested by Professor Cayley in reply to a re- quest of the writer. The word originated in the following construction for the surface, which was also communicated by Professor Cayley. Cut the cylinder cc* +y* = (pp p a }* in an ellipse by the plane z = x, and consider the line x = o, y p$ p a . If any plane z = c cuts the ellipse in the points A, B and the line in (7, then CA, CB are two generating lines of the surface. THE CYLINDROID. 17 sin ,4 A + A = A + A ~ ^ cot -A) cot A + with similar values for / and 22- It is therefore obvious that the cylindroid is determined, and that the solution is unique. It will often be convenient to denote by (0, 0) the cylindroid drawn through the two screws and <. 17. General Property. The general property of the cylindroid, which is of importance for our present purpose, may be thus stated. If a body receive twists about three screws on a cylindroid, and if the amplitude of each twist be proportional to the sine of the angle between the two non-corresponding screws, then the body after the last twist will occupy the same position w r hich it did before the first. The proof of this theorem must, according to 15, involve the proof of the following : If a body be acted upon by wrenches about three screws on a cylindroid, and if the intensity of each wrench be proportional to the sine of the angle between the two non-corre- sponding screws, then the three wrenches equilibrate. The former of these properties of the cylindroid is thus proved : Take any three screws 9, 0, i/>, upon the surface w T hich make angles /, m, n, with the axis of x, and let the body receive twists about these screws of amplitudes 0', $', ;//. Each of these twists can be de- composed into two twists about the screws a and )3 which lie along the axes of x and y. The entire effect of the three twists is, therefore, reduced to two rotations around the axes of x and y, and two translations parallel to these axes. c 1 8 THE CYLINDROID. The rotations are through angles equal respectively to / cos / + 0' cos m + i// cos n and Q' sin / + 0' sin m + ;// sin n. The translations are through distances equal to p a (&' cos / + 0' cos m + T//COS n) and pft (0' sin / + $' sin ?;/ + $' sin 72). These four quantities vanish if & X V sin (m - n) sin (/z - /) sin (/ - m) 9 and hence the fundamental property of the cylindroid lias been proved. The cylindroid affords the means of compounding two twists (or two wrenches) by a rule as simple as that which the parallelogram of force provides for the com- position of two intersecting forces. Draw the cylindroid which contains the two screws ; select the screw on the cylindroid which makes angles with the given screws whose sines are in the inverse ratio of the amplitudes of the twists (or the intensities of the wrenches) ; a twist (or wrench) about the screw so determined is the required resultant. The amplitude of the resultant twist (or the intensity of the resultant wrench) is proportional to the diagonal of a parallelogram of which the two sides are parallel to the given screws, and of lengths proportional to the given amplitudes (or intensities). 18. Particular Cases. If/ a = pp the cylindroid re- duces to a plane, and the pitches of all the screws are equal. If the pitches be all zero, then the general pro- perty of the cylindroid reduces to the well known con- struction for the resultant of two intersecting forces, or of rotations about two intersecting axes. If the pitches THE CYLINDROID. 19 be all infinite, the general property reduces to the con- struction for the composition of two translations or of two couples. 19. Form of the Cylindroid. The equation of the surface only contains the single parameter pa. pp y consequently all cylindroids are similar surfaces only differing in absolute magnitude. The curved portion of the surface is contained be- tween the two parallel planes z = + (p a - pp} y but it is to be observed that the nodal line x = o, y = o, also lies upon the surface. The intersection of the nodal line with a plane is a double point (connode) or a conjugate point (acnode) upon the curve in which the plane is cut by the cylin- droid according as the point does lie or does not lie between the two bounding planes. A model of a portion of the cylindroid is represented in the frontispiece. In order to realize from the model the actual form of the surface, the diameter of the central cylinder must be conceived to be evanescent, and the radiating wires must be extended to infinity. 20. The Pitch Conic. Besides being acquainted with the form of the cylindroid, it is also very useful to have a clear view of the distribution of pitch upon the screws contained on the surface. The surface being given, one arbitrary element must be further specified before that distribution is known. If, however, two screws be given, then both the surface and the distribution are determined. Any constant quantity may be added to all the pitches of a certain distribution, and the distribu- tion thus modified is still a possible one. Let p e be the pitch of a screw on the cylindroid which makes an angle / with the axis of x\ then ( 1 1) p9 = fa cos 2 / + p ft sin'V. C 2 20 THE CYLINDROID. Draw in the plane #, y, the conic where H is any constant ; and if r be the radius vector which makes an angle / with the axis of x, we have &*?:> whence the pitch of each screw on a cylindroid is pro- portional to the inverse square of the parallel diameter of the pitch conic. Being given the cylindroid, we require further to know the eccentricity of the pitch conic, and then the pitches of all the screws are determined. 21. Summary. It is one of the main objects of the present essay to associate a geometrical conception with the solution of each problem. To do this effectively we- shall often have occasion to make use of the principle demonstrated in this chapter, viz., That a cylindroid can be drawn so that not only shall two of its generators coincide with any two given screws a and j3, but that when all the generators of the surface become screws by having pitches assigned to them according to the law of distribution enunciated in 20, the pitches assigned to the generators which coincide with a and ]3 shall be equal to the given pitches of a and ]3. ( 21 ) CHAPTER III. RECIPROCAL SCREWS. 22. Reciprocal Screws. If a body only free to twist about a screw a be in equilibrium, though acted upon by a wrench on the screw /3, then conversely a body only free to twist about the screw j3 will be in equili- brium, though acted upon by a wrench on the screw a. The principle of virtual velocities states, that if the body be in equilibrium the work done in a small dis- placement against the external forces must be zero ; but the condition for this is, that the virtual coefficient should vanish (13), or (f a + f ft ) cos O - d sin O = o. The symmetry shows that precisely the same condi- tion is required whether the body be free to twist about , while the wrench act on j3, or vice versa. A pair of screws are said to be reciprocal when their virtual coeffi- cient is zero. 23. Particular Instances. Parallel or intersecting screws are reciprocal when the sum of their pitches is ^ero. Screws at right angles are reciprocal either when they intersect, or when one of the pitches is infinite. Two screws of infinite pitch are reciprocal, because a couple could not move a body which was only susceptible of translation. A screw whose pitch is zero or infinite is reciprocal to itself.* * For other particular instances see Professor Everett " On a New Method in Statics and Kinematics," 27 ; " Messenger of Mathematics," New Series, JS T o. 39, 1874. 22 RECIPROCAL SCREWS. 24. Screw Reciprocal to Cylindroid. If a screw T? be- reciprocal to two given screws 6 and 0, then rj is recipro- cal to every screw on the cylindroid (0, 0). For a body only free to twist about rj would be undis- turbed by wrenches on 9 and ; but a wrench on any screw i// of the cylindroid can be resolved into wrenches on 9 and ; therefore a wrench on i/> cannot disturb a body only free to twist about rj; therefore i// and rj are reciprocal. We may say for brevity that rj is reciprocal to the cylindroid. TJ cuts the cylindroid in three points,* and one screw of the cylindroid passes through each of these three points ; these three screws must, of course, be reciprocal to 17. Now two intersecting screws can only be reciprocal when they are at right angles, or when the sum of their pitches is zero. The pitch of the screw upon the cylin- droid which makes an angle / with the axis of x is p a cos 2 / + pp sin 2 /. This is also the pitch of the screw TT - I. There are, therefore, two screws of any given pitch ; but there can- not be more than two. It follows that TJ can at most in- tersect two screws upon the cylindroid of pitch equal and opposite to its own ; and, therefore, r\ must be perpendi- cular to the third screw.f Hence any screw reciprocal to a cylindroid must intersect one of the generators at right angles. We easily infer, also, that a line intersect- ing one screw of a cylindroid at right angles, must cut the surface again in two points, the screws passing through which have equal pitch. 25. Reciprocal Cone. From any point P perpen- * Every right line meets a surface of the third degree in three points.. Salmon, "Analytic Geometry of Three Dimensions," 2nd Ed., p. 14. + The writer may, perhaps, be excused for adding that it was the percep- tion of this point which first gave him clear views on the subject of the present volume. RECIPROCAL SCREWS. diculars can be let fall upon the generators of the cylindroid, and if to these perpendiculars pitches are assigned which are equal in magnitude and opposite in sign to the pitches of the two remaining screws on the cylindroid intersected by the perpendicular, then the perpendiculars form a cone of reciprocal screws. We shall now prove that this cone is of the second order, and we shall show how it can be constructed. Let O be the point from which the cone is to be drawn, and through O let a line 07" be drawn which is parallel to the nodal line, and, therefore, perpendicular to all the generators. This line will cut the cylindroid in one real point T (Fig. 2), the two other points of inter- section coalescing into the infinitely distant point in which OT intersects the nodal line. Draw a plane through T and through the screw L M which, lying on the cylin- droid, has the same pitch as the screw through T. Now this plane must cut the cylindroid in a conic section, for the line LM and the conic will then make up the curve of the third degree, in which the plane must cut the sur- face.* Also since the entire cylindroid (or at least its curved portion) is included between two parallel planes, 19, it follows, that this conic must be an ellipse. We shall now prove that Fig. 2. Salmon, "Analytic Geometry of Three Dimensions," 2nd Ed., p. 14. 24 RECIPROCAL SCREWS. this ellipse is the locus of the feet of the perpendiculars let fall from O on the generators of the cylindroid. Draw in the plane of the ellipse any line TUV through T; then, since this line intersects two screws of equal pitch in T and U, it must be perpendicular to that gene- rator of the cylindroid which it meets at V. This generator is, therefore, perpendicular to the plane of OT and VT y and, therefore, to the line O V. It follows that V must be the foot of the perpendicular from O on the generator through V y and that, therefore, the cone drawn from O to the ellipse TL VM is the cone required. We hence deduce the following construction for the cone of reciprocal screws which can be drawn to a cylin- droid from any point O. Draw through O a line parallel to the nodal line of the cylindroid, and let T be the one real point in which this line cuts the surface. Find the second screw L M on the cylindroid which has a pitch equal to the pitch of the screw which passes through T. A plane drawn through the point T and the straight line L M will cut the cylindroid in an ellipse, the various points of which joined to O give the cone required. We may further remark that as the plane TLM passes through a generator it must be a tangent plane to the cylindroid at the point Z, while at the point M the line LM must intersect another generator.* It follows that L must be the foot of the perpendicular from Tupon L M y and that M must be a point upon the nodal line. 26. Locus of a Screw Reciprocal to Four Screws. Since a screw is determined by five quantities, it is clear that when the four conditions of .reciprocity are fulfilled the screw must be confined to a certain ruled surface. Now this surface can be no other than a cylin- * Salmon, "Analytic Geometry of Three Dimensions," 2nd Ed., p. 348. RECIPROCAL SCREWS. 25 droid. For, suppose that three screws A, ju, v, which were reciprocal to the four given screws did not lie on the same cylindroid, then any screw on the cylindroid (A, ju), and any screw i// on the cylindroid (A, v) must also fulfil the conditions, and so must also every screw on the cylindroid (0, \/,) ( 4). We should thus have the screws reciprocal to four given screws, limited not to one surface, but to a family of surfaces, which is impossible. The construction of the cylindroid which is the locus of all the screws re- ciprocal to four given screws, may be effected in the fol- lowing manner : Let a, |3, 7, S be the four screws, of which the pitches are in descending order of magnitude. Draw the cylin- droids (a, 7) and Q3, ). If or be a linear magnitude inter- mediate between p ft and / 7 , it will be possible to choose two screws of pitch of the determination of a screw reciprocal to five given screws must admit of a finite number of solutions, because the number of conditions to be fulfilled is the same as the number of disposable constants. Now it is very important to observe that that number must be one. For if two screws could be found which fulfilled the neces- * Two lines can be drawn which will intersect four non-intersecting lines. -Salmon, "Analytic Geometry of Three Dimensions," 2nd Ed., page 426. 26 RECIPROCAL SCREWS. sary conditions, then these conditions would be equally fulfilled by every screw on the cylindroid determined by those screws ( 24), and therefore the number of solutions of the problem would not be finite. The construction of the screw whose existence is thus demonstrated, can be effected by the results of the last article. Take any four of the five screws, and draw the reciprocal cylindroid which must contain the required screw. Any other set of four will give a different cylin- droid, which also contains the required screw. These cylindroids must therefore intersect in the single screw,, which is reciprocal to the five given screws. 28. Screw upon a Cylindroid Reciprocal to a Given Screw. Let e be the given screw, and let X, ju, v, p be any four screws reciprocal to the cylindroid ; then the single screw TJ, which is reciprocal to the five screws t, A, /i, i, p y must lie on the cylindroid because it is reciprocal to A, ju, v y p, and therefore r\ is the screw required. The solution must be unique, for if a second screw were reciprocal to f, then the whole cylindroid would be- reciprocal to c ; but this is not the case unless i fulfil cer- tain conditions (24). 29. Properties of the Cylindroid. We add here a few properties of the cylindroid for which the writer is prin- cipally indebted to his friend Dr. Casey. The ellipse in which a tangent plane cuts the cylin- droid has a circle for its projection on a plane perpendi- cular to the nodal line, and the radius of the circle is the minor axis of the ellipse. The difference of the squares of the axes of the ellipse is constant 'wherever the tangent plane be situated. The minor axes of all the ellipses lie in the same plane. The line joining the points in which the ellipse is cut by two screws of equal pitch on the cylindroid is parallel to the major axis. RECIPROCAL SCREWS. 27 The line joining, the points in which the ellipse is cut by two intersecting screws on the cylindroid, is pa- rallel to the minor axis. All the screws which lie in a plane and are reciprocal to a cylindroid envelope a conic. The relation of this conic to the cubic in which the plane cuts the cylindroid might have some geometrical interest. As we have no physical applications to make of these theorems, the demonstrations are not given. CHAPTER IV. SCREW CO-ORDINATES. 30. Introduction. We are accustomed, in ordinary sta- tics, to resolve the forces acting on a rigid body into three forces acting along given directions at a point and three couples in three given planes. In the present theory we are, however, led to regard a force as a wrench on a screw, of which the pitch is zero, and a couple as a wrench on a screw of which the pitch is infinite. The familiar process just referred to is, therefore, only a special case of the more general method of resolution by which the intensities of the six wrenches on six given screws can be determined, so that, when these wrenches are compounded together, they shall constitute a wrench of given intensity on a given screw.* The problem which has to be solved may be stated in a more symmetrical manner as follows : To determine the intensities of the seven wrenches on seven given screws, such that, when these wrenches are applied to a rigid body, which is entirely free to move in any way, they shall equilibrate. The solution of this problem is identical (15) with that which may be enunciated as follows : To determine the amplitudes of seven small twists about seven given screws, such that, if these twists be * If all the pitches be zero, the problem stated above reduces to the deter- mination of the six forces along six given lines which shall be equivalent to a given force. If further, the six lines of reference form the edges of a tetrahe- dron, we have a problem which has been solved by Mobius, Crelle's Journal, t. xviii., p. 207. SCREW CO-ORDINATES. 29 applied to a rigid body in succession, the body after the last twist shall occupy the same position which it had before the first. The problem w r e have last stated has been limited as usual to the case where the amplitudes of the twists are small quantities, so that the motion of each particle produced by each twist is sensibly rectilinear. Were it not for this condition a distinct solution would be re- quired for every variation of the order in which the suc- cessive twists were imparted. If the number of screws were greater than seven, then both problems would be indeterminate ; if the num- ber were less than seven, then both problems would be impossible (unless the screws were specially related) ; the number of screws being seven, the problem of the determination of the ratios of the seven intensities (or amplitudes) has, in general, one solution. We shall solve this for the case of wrenches. Let the seven screws be a, /3, 7, S, e, , rj. Find the screw \p which is reciprocal to y, 3, e, , rj. Let the seven wrenches act upon a body only free to twist about i//. The reaction of the constraints which limit the motion of the body will neutralize every wrench on a screw re- ciprocal to ^ ( 22). We may, therefore, so far as a body thus circumstanced is concerned, discard all the wrenches except those on a and /3. Draw the cylindroid (a, /3), and determine thereon the screw p which is reciprocal to i//. The body will not be in equilibrium unless the wrenches about a and j3 constitute a wrench on /o, and hence the ratio of the intensities a." and $" is determined. By a similar process the ratio of the intensities of the wrenches on any other pair of the seven screws may be determined, and thus the problem has been solved. 31. Intensities of the Components. Let the six screws of reference be w,, &c. w 6 , and let p be a given screw 30 SCREW CO-ORDINATES. on which is a wrench of given intensity /o". Let the intensities of the components be p/', &c. p 6 /x , and let i\ be any screw. By the principle of 10, a twist about rj must do the same quantity of work acting directly against the wrench on p as the sum of the six quantities of work which would be done by the same twist against each of the six components of the wrench on p. We, therefore, have the equation (using the notation of p. 13) P // ^np= P"*^ + &C. + pe'X-.- By taking five other screws in place of 77, five more equations are obtained, and from the six equations thus found p/', &c. p 6 " can be determined. This process will be greatly simplified by judicious choice of the six screws of which r/ is the type. Let j/ be reciprocal to w- 2 , &c. w 6 , then *r nua = o &c. ^- na)6 = o, and we have f >// ^ P = p/'w,,^. From this equation pi" is at once determined, and by five similar equations the intensities of the five remaining components may be likewise found. Precisely similar is the investigation which deter- mines the amplitudes of the six twists about the six screws of reference into which any given twist may be decomposed. 32. The Intensity of the Resultant may be expressed in terms of the intensities of its components on the six screws of reference. Let p be any screw of pitch / p , and let p l9 &c. p Q be the pitches of the six screws of reference w b &c. w 6 ; then taking for i\ in ( 26), each of the screws of reference in succession, and remembering that the virtual coefficient of two coincident screws is simply double the pitch, we have the following equations : P // ^ pWi = p/^j + p 2 // ^ ft , i(02 -f &C. + p 6 "a-- a 6 &C. = &C. SCREW CO-ORDINATES. 31 But taking the screw p in place of 17 we have Substituting for sr p(0i &c. TOpWo from the former equa- tions, we deduce A p" This result may recall the well-known expression for the square of a force acting at a point in terms of its components along three axes passing through the point. This expression is greatly simplified when the three axes are rectangular, and we shall now show that by a special disposition of the screws of reference, a corresponding simplification can be made in the formula just written. 33- Co-Reciprocal Screws. We have hitherto chosen the six screws of reference quite arbitrarily; we now proceed in a different manner. Take for o>,, any screw ; for w 2 any screw reciprocal to wi ; for a> 3 , any screw reciprocal to ui and w 2 ; for w iy any screw reciprocal to w,, 2 , w 3 ; for w 5 , any screw reciprocal to wi, w 2 , w 3 , w 4 ; for 72, 73> 74, 75* 76> The relative values of pi, &c., p 6 , being thus found, the absolute values are given by 37. The condition that six screws have a common recipro- cal screw is expressed by the evanescence of a deter- minant, which may be compared with the condition that three straight lines be coplanar, of which the direction cosines are given. 42. Co-ordinates of a Screw on a Cylindroid. We may define the screw on the cylindroid by the angle /, which it makes with the screw a on the axis of x. Since a wrench of unit intensity on has components of intensities cos / and sin / on a and j3 ( 1 7), and since each of these components may be resolved into six wrenches on any six co-reciprocal screws, we must have ( 36) O n = a n COS / + j3 n sin /. From this expression we can find the pitch of 6 : for we have p e = S/i (ai cos / + ]3i sin If whence expanding and observing that as a and ]3 are re- ciprocal 2^ T aij3i = and also that ^p l a^=p a and S^ t j3i 2 = pp, we have the expression already given ( 20) viz. = COS SCREW CO-ORDINATES. 37 If two screws, 6 and 0, upon the cylindroid, are reci- procal, then (m being the defining angle of 0), 2/i (ai cos / + j3i sin /) ( i) : for suppose the body receive any n twists about A i, &c., A n the position attained could have been reached by a twist about some single screw A . It follows that the body must be free to twist about A . Now since the amplitudes of the n twists may have any magnitude (each not exceeding an infinitely small quantity), A is merely one of an infinite number of screws, about which twist- ing must be possible. All these screws, together with A &c., A , we call a screw complex of the n ih order. If it be found that the body cannot be twisted about any screw which does not belong to the screw complex of the n th order, then the body is said to have freedom of the n th order. It may be necessary to remark that A &c., A ny must not be themselves members of a screw complex of order lower than n. If this were the case, the screws EQUILIBRIUM OF A RIGID BODY. 39 about which the body could be twisted would only con- sist of the members of that lower screw complex. Since the amplitudes of the n twists about A . . . A n are [arbitrary, it might be thought that there are n dis- posable quantities in the selection of a screw S from a screw complex of the n th order. It is, however, obvious from 1 7 that the determination of the position and pitch of S depends only upon the ratios of the amplitudes of the twists about A lf . . . A n and hence in the selection of a screw from the screw complex of the n th order, we have n - i disposable quantities. 44. Constraints. An essential feature of a system of constraints consists in the number of independent quantities which are necessary to specify the position of the body when displaced in conformity with the require- ments of the constraints. That number which cannot be less than one, nor greater than six, is the order of the freedom. To each of the six orders of freedom a certain type of screw complex is appropriate. The study of the six types of screw complex as here defined is a problem of kinematics, but the statical and kinematical properties of screws are so interwoven that we derive great advantages by not attempting to rele- gate the statics and kinematics to different chapters. We shall not require any further mention of the con- straints. Every conceivable condition of constraints must have been included when the six screw complexes are discussed in their most general form. Nor does it come within our scope, except on rare occasions, to specialize the enunciation of any problem, further than by men- tioning the order of the freedom permitted to the body. 45. Screw Reciprocal to a Complex. If a screw X be reciprocal to n screws, A l9 &c., A, belonging to a screw complex of order n, then X is reciprocal to every other screw A which belongs to the same screw com- 40 EQUILIBRIUM OF A RIGID BODY. plex. For, by the definition of the screw complex, it ap- pears that twists of appropriate amplitudes about A ,, &c., A n , would compound into a twist about A. It fol- lows ( 43) that wrenches on AU &c., A n , of appropriate intensities (32) compound into a wrench on A . Suppose these wrenches on A\ y &c., A n , were applied to a body only free to twist about X, then since X is reciprocal to A i, &c., A n , the equilibrium of the body would be un- disturbed. The resultant wrench on A must therefore be incapable of moving the body, therefore^ and X must be reciprocal. 46. The Reciprocal Screw Complex. All the screws which are reciprocal to a screw complex P of order k constitute a screw complex Q of order 6 - k. This im- portant theorem is thus proved : Since only one condition is necessary for a pair of screws to be reciprocal, it follows, from the last section, that if a screw X be reciprocal to P it will fulfil k con- ditions. The screw X has, therefore, 5 - k elements still disposable, and consequently (k < 5) an infinite number of screws Q can be found which are reciprocal to the screw complex P. The theory of reciprocal screws will now prove that Q must really be a screw complex of order 6 - k. In the first place it is manifest that Q must be a screw complex of some order, for, in general, if a body be capable of twisting about even six screws, it must be perfectly free. Here, however, if a body were able to twist about the infinite number of screws em- bodied in Q y it would still not be free, because it would remain in equilibrium, though acted upon by a wrench about any screw of P. If follows that Q can only denote the collection of screws about which a body can twist which has some definite order of freedom. It is easily seen that that number must be 6 - k, for the number of constants disposable in the selection of a screw belong- EQUILIBRIUM OF A RIGID BODY. 41 ing to a screw complex is one less than the order of the complex (38). But. we have seen that the constants dis- posable in the selection of X are 5 - k, and, therefore, Q must be a screw complex of order 6 - k. We thus see, that to any screw complex P of order k be- longs a reciprocal screw complex Q of order 6 - k. Every screw of P is reciprocal to all the screws of Q, and vice versa. This theorem provides us with a definite test as to whether any given screw a. is a member of the screw complex P. Construct any 6 - k screws of the reciprocal system. If then a be reciprocal to these 6 - k screws, a must belong to P. We thus have 6 - k conditions to be satisfied by any screw when a member of a screw com- plex of order k. 47. Equilibrium. If the screw complex P expresses the freedom of a rigid body, then the body will remain in equilibrium though acted upon by a wrench on any screw of the reciprocal screw complex Q. This is, perhaps, the most general theorem which can be enunciated with respect to the equilibrium of a rigid body. This theorem is thus proved : Suppose a wrench to act on a screw rj belonging to Q. If the body does not continue at rest, let it commence to twist about a. We thus have a wrench about TJ disturbing a body which twists about a, but this is impossible, because a and j are reciprocal. In the same manner it may be shown that a body which is free to twist about all the screws of Q will not be disturbed by a wrench about any screw of P. Thus, of two reciprocal screw complexes, each expresses the locus of a wrench which is unable to disturb a body free to twist about any screw of the other. 48. Reaction of Constraints. It also follows that the reactions of the constraints by which the movements of a body are confined to twists about the screws of 42 EQUILIBRIUM OF A RIGID BODY. a complex P can only be wrenches on the reciprocal screw complex Q, for the reactions of the constraints ar& only manifested by the success with which they resist the efforts of certain wrenches to disturb the equilibrium of the body. 49. Parameters of a Screw Complex. We next con- sider the question as to how many parameters are required in order to specify completely a screw com- plex of the n th order. Since the complex is defined when n screws are given, and since five data are required for each screw, it might be thought that $n parameters would be necessary. It must be observed, however, that the given $n data suffice not only for the purpose of de- fining the screw complex but also for pointing out n special screws upon the screw complex, and as the point- ing out of each screw on the complex requires n i quantities ( 38), it follows that the number of parameters, actually required to define the complex is only $n- n(n- i) = n (6 - n). This result has a very significant meaning in con- nexion with the theory of reciprocal screw complexes P and Q. Assuming that the order of P is n, the order of Q is 6 - n ; but the expression n (6 -n ) is unaltered by changing n into 6 - n. It follows that the number of parameters necessary to specify a screw complex is identical with the number necessary to specify its reci- procal screw complex. This remark is chiefly of impor- tance in connexion with the complexes of the fourth and fifth orders, which are respectively the reciprocal com- plexes of a cylindroid and a single screw. We are now assured that a collection of all the screws which are re- ciprocal to an arbitrary cylindroid can be nothing less than a screw complex of the fourth order in its most general type, and also, that all the screws in space which EQUILIBRIUM OF A RIGID BODY. 43, are reciprocal to a single screw must form the most general type of a screw complex of the fifth order. 50. Applications of Co-ordinates. If the co-ordinates of a screw satisfy n linear equations, the screw must belong to a screw complex of the order 6 - n. Let t\ be the screw, and let one of the equations be AM +, &c., + A MS = o, whence ?? must be reciprocal to the screw whose co-ordi- nates are proportional to It follows that TJ must be reciprocal to n screws, and therefore belong to a screw complex of order 6 - n. Let cr, j3, 7, be four screws about which a body re- ceives twists of amplitudes a, /3 X , y', S'. It is required to find the co-ordinates of the screw p and the amplitude p' of a twist about p which will produce the same effect as the four given twists. Wehaveseen (39) that the twist about any screw a, may be resolved in one way into six twists of amplitudes a'cti, . . . a'ae, on the six screws of reference ; we must therefore have p'pi = of en + |3' )3i + y ji + ' Si &c., &c. p'.pe = a a 6 + j3' j3 6 + y' 7e + 8' S 6 whence p' and p l9 . . . p B can be found (37). A similar process will determine the co-ordinates of the resultant of any number of twists, and it follows from 1 5 that the resultant of any number of wrenches is to be found by equations of the same form. In ordinary mechanics, the conditions of equilibrium of any number of forces are six, viz., that each of the three forces, and each of the three couples to which the system is equivalent shall vanish. In the present theory the conditions are 44 EQUILIBRIUM OF A RIGID BODY. likewise six, viz., that the intensity of each of the six wrenches on the screws of reference to which the given system is equivalent shall be zero. Any screw will belong to a complex of the n th order if it be reciprocal to 6 - n independent screws ; it follows that 6 - n conditions must be fulfilled when n -t- i screws belong to a screw complex of the n th order. To determine these conditions we take the case of n = 3, though the process is obviously general. Let a, /3, 7, S be the four screws, then since twists of amplitudes '> j3'> 7', $' neutralise, we must have p' zero and hence the six equations 7 / 7l + S'& = o, &c. from any four of these equations the quantities a' y /3', 7', $ can be eliminated, and the result will be one of the required conditions. It is noticeable that the 6 - n conditions are often presented in the evanescence of a single function, just as the evanescence of the sine of an angle between a pair of straight lines embodies the two conditions necessary that the direction cosines of the lines coincide. The function is suggested by the following considerations : If n + 2 screws belong to a screw complex of the (n + i) th order, twists of appropriate amplitudes about the screws neutralise. The amplitude of the twist about any one screw must be proportional to a function of the co-ordi- nates of all the other screws ; this is evident, because if one amplitude were ascertained to be zero, the remaining screws must belong to a complex of the n th order. We thus see that the evanescence of one function must afford all that is necessary for n + i screws to belong to a screw complex of the n 1h order. 45 CHAPTER VI. THE PRINCIPAL SCREWS OF INERTIA. 51. Introduction. If a rigid body be free to rotate about a fixed point, then it is well known that an impulsive couple in a plane perpendicular to one of the principal axes which can be drawn through the point will make the body commence to rotate about that axis. Suppose that on one of the principal axes lay a screw rj with a very small pitch, then a twisting motion about rj would closely resemble a simple rotation about the correspond- ing axis. An impulsive wrench on ij will, when united with the reaction of the fixed point, reduce to a couple in a plane perpendicular to the axis. If we now sup- pose the pitch of TJ to be evanescent, we may still assert that an impulsive wrench on TJ of very great in- tensity will cause the body, if previously quiescent, to commence to twist about rj. We have stated a familiar property of the principal axes in this indirect manner, for the purpose of showing that it is merely an extreme case fora body with freedom of the third order of the following general theorem : If a quiescent rigid body have freedom of the n th order, then n screws can always be found (but not more than n\ such that if the body receive an impulsive wrench on any one of these screws, the body will commence to twist about the same screw. These n screws are of great significance in the pre- sent method of studying Dynamics, and they may be termed the principal screws of inertia. In the present chapter we shall prove the general theorem just stated, 46 THE PRINCIPAL SCREWS OF INERTIA. while in the chapters on the special orders of freedom we shall show how the principal screws of inertia are to be determined for each case. 52. Screws of Reference. We have now to define the group of six co-reciprocal screws ( 2 8) which are pecu- liarly adapted to serve as the screws of reference in Kinetic investigations. Let O be the centre of inertia of the rigid body, and let OA, OB, OC be the three prin- cipal axes through O, while a, b, c are the corresponding radii of gyration. Then two screws along OA, viz. : w : , W2, with pitches + a, - a ; two screws along OB, viz. : o> 3 , <>4, with pitches + b, - b, and two along OC, viz. : o> 5 , W G , with pitches + c, - c, are the co-reciprocal group which we shall employ. For convenience in writing the formulae, we shall often use / &c. p Q , to denote the pitches as before. We shall now prove that the six screws thus defined are the principal screws of inertia of a free rigid body. Let the mass of the body be M, and let an impulsive wrench on wi act for a short time /. The intensity of this wrench is tui", and the moment of the couple is a^ 1 . We now consider the effect of the two portions of the wrench separately. The effect of the force wi" is to give the body a velocity of translation parallel to OA and equal to -^. wi ". By the property of the principal axes the effect of the couple will be to impart an angular ve- locity w/ about the axis OA. This angular velocity is easily determined. The effective force which must have acted upon a particle dm at a perpendicular dis- tance r from OA is -~ dm. The sum of the moments of all these forces is Ma? ^ . This quantity must equal THE PRINCIPAL SCREWS OF INERTIA. 47 the moment of the given couple or / whence i)\ = ^ i . aM The effect of an impulsive wrench on wi is, therefore, to give the body a velocity of translation parallel to OA, and equal to -r- f w/', and also a velocity of rotation about OA equal to ^-, to/'. These movements unite to form a aM twisting motion about a screw on OA, of which the pitch is found by dividing the velocity of translation by the velocity of rotation to be equal to a, this being the pitch of wi, proves that an impulsive wrench on wi will make the body commence to twist about wi, and that, therefore, wi is a principal screw of inertia. Similar reasoning applies to the remaining five screws. 53. Impulsive Screws and Instantaneous Screws. If a free quiescent rigid body receive an impulsive wrench on a screw rj, the body will immediately commence to twist about an instantaneous screw a. The co-ordinates of a being given, it is required to determine the co- ordinates of jj. The impulsive wrench on ?> of intensity r\" is to be decomposed into components of intensities r\ f/ in, &c. V'lje on wi, &c w 6 . The component on w will generate a twist velocity about w amounting to but if 'a! be the twist velocity about a which is finally produced, the expression just written must be equal to ' a n , and hence we have the following useful result : If the co-ordinates of the instantaneous screw be proper- 48 THE PRINCIPAL SCREWS OF INERTIA. tional to ai, &c. a 6 , then the co-ordinates of the correspond- ing impulsive screw are proportional to p l ai, &c. / 6 a 6 . 54. Conjugate Screws of Inertia. If a and j3 be two instantaneous screws, and if r\ and be the correspond- ing impulsive screws, then when a is reciprocal to % we must have /3 reciprocal to rj. We shall first suppose the body to be perfectly free. The co-ordinates of S are proportional to/i/3i, &c./ 6 j3 6 , hence the condition that a and are reciprocal ( 34) is /i 2 ai |3i + &c. + / 6 2 a 6 /3 6 = o. But this is precisely the equation which we should have found by expressing the condition that ]3 and rj were reciprocal. When this relation is fulfilled, the screws a and ]3 are said to be conjugate screws of inertia. We shall now show that this theorem will still remain true even if the body be only partially free. When the body receives an impulsive wrench on there is an im- pulsive reaction of the constraints on a screw ju . The effect on the body is, therefore, the same as if it had been free, but had received an impulsive wrench of which the component on o>i had the intensity /x i + ju"/^ ; hence, h being a constant, we have &c. &c. ", + juV multiplying the first of these equations by p l a\, the second by/ 2 2, &c., adding the six products, and re- membering that a and are reciprocal by hypothesis, while a and /it are reciprocal, by the nature of the re- actions of the constraints ( 43), we have, as before THE PRINCIPAL SCREWS OF INERTIA. 49 Precisely the same condition must be satisfied when ]3 and ij are reciprocal, and hence the general property of con- jugate screws of inertia is true, whether the body be free or constrained in any way. 55. The Determination of the Impulsive Screw, corres- ponding to a given instantaneous screw, is a definite problem when the body is perfectly free. If, however, the body be constrained, we shall show that any screw selected from a certain screw complex will fulfil the re- quired condition. Let B l9 &c. jB 6 _ n be 6 - n screws selected from the screw complex which is reciprocal to that corresponding to the freedom of the n th order possessed by the rigid body. Let S be the screw about which the body is to twist. Let X be any screw, an impulsive wrench about which would make the body twist about S; then any screw Y belonging to the screw-complex of the (7 - n} th order, specified by the screws, X, B ly &c. -Z? 6 . n is an im- pulsive screw, corresponding to S as an instantaneous screw. For the wrench on Y may be resolved into 7-72 wrenches on X, B\ y &c. -# 6 -n ; of these, all but the first are instantly destroyed by the reaction of the constraints, so that the wrench on Kis practically equivalent to the wrench on X, which, by hypothesis, will make the body twist about S. For example, if the body had freedom of the fifth order, then an impulsive wrench on any screw on a cer- tain cylindroid will make the body commence to twist about a given screw. If a body have freedom of the third order, then the "locus" of an impulsive wrench which would make the body twist about a given screw consists of all the screws in space which are reciprocal to a certain cylindroid. 56. System of Conjugate Screws of Inertia. We shall now show that from the screw-complex of the n th order P, E 50 THE PRINCIPAL SCREWS OF INERTIA. which expresses the freedom of the rigid body, n "screws can be selected so that every pair of them are conjugate screws of inertia (54). Let B ly &c. 7? 6 _ n be (6 - n) screws defining the reciprocal screw-complex. Let A i be any screw belonging to P. Then in the choice of A A l we have n - i arbitrary quantities. Let /i be any impulsive screw corresponding to A v as an instantaneous screw. Choose A z reciprocal to /i, B ly &c. 7? 6 _ n , then AI and A 2 are conjugate screws, and in the choice of the latter we have n- 2 arbitrary quantities. Let 7 2 be any impulsive screw corresponding to A 2 as an instantaneous screw. Choose A 3 reciprocal to I l9 7 2 , B l9 &c. 7? 6 _ M , and proceed thus until A n has been attained, then each pair of the group AU &c. A n are conjugate screws of inertia. The number of quantities which remain arbitrary in the choice of such a group amount to n (n - i) n 1+72 2 + &C. + I = ', 2 or exactly half the total number of arbitrary constants in the selection of any n screws from a complex of the n th order. 57. Principal Screws of Inertia. It is the object of this section to show that it is always possible to select from the screw-complex of the n th order expressing the freedom of a rigid body, one group of n screws, of which every pair are both conjugate and reciprocal, and that these constitute the principal screws of inertia ( 51). To prove this, it is sufficient to show that when the remaining half of the arbitrary constants ( 56) have been suitably disposed, then the group of n screws be- sides being conjugate will be co-reciprocal. Choose A\ reciprocal to B\ y &c. 7? 6 _ n , with n - i arbitrary quantities ; A 2 reciprocal to Ai 9 B ly &c. B n -\, with n - 2 arbitrary quantities, and so on, then the total number of arbitrary THE PRINCIPAL SCREWS OF INERTIA. 5 1 quantities in the choice of n co-reciprocal screws from a complex of the n lh order is n(n- i) n i+n z...+ i= 2 Hence, by suitable disposition of the n(n - i) constants we can find one group of n screws which are both con- jugate and co-reciprocal. We have now to show that these n screws are really the principal screws of inertia (51). yWe shall state the argument for the freedom of the third order, the argu- ment for any other order being precisely similar. Let A i, A 2, A z , be the three conjugate and co-reci- procal screws which can be selected from a complex of the third order. Let B^B^ Bj, be any three screws belong- ing to the reciprocal screw-complex. Let R l9 jR 2 , R* be any three impulsive screws corresponding respectively to A l9 A 2 , A 3 as instantaneous screws. An impulsive wrench on any screw belonging to the screw-complex of the 4* order defined by Ri 9 B l9 B 2 , B* will make the body twist about A\ (55), but the screws of such a complex are reciprocal to A z and A 3 ; for since A i and A z are conjugate, ^ must be reciprocal to A z ( 54), and also to A 3 , since AI and Az are conjugate. It follows from this that an impulsive wrench on any screw reciprocal to A z and A 3 will make the body commence to twist about A 19 but A 1 is itself reciprocal to A z and A 3 , and hence an impulsive wrench on AI will make the body commence to twist about AI. Hence AI and also At and A 3 are principal screws of inertia. We shall now show that with the exception of the n screws here determined, no other screw possesses the property in question. Suppose another screw S were to possess this property. Decompose the wrench on S into n wrenches of intensities Si", &c. S n "onA lf &c. A, this must be possible, because if the body is to be capable of E 2 5 2 THE PRINCIPAL SCREWS OF INERTIA. twisting about S this screw must belong to the complex specified by A Jy &c. A n . The n impulsive wrenches on AI, &c. A n will produce twisting motions about the same screws, but these twisting motions are to compound into a twisting motion on S. It follows that the component twist velocities Si, &c. S n ' must be proportional to the intensities Si", &c. S n ". But if this were the case, then every screw of the complex would be a principal screw of inertia ; for let X be any impulsive screw, and suppose that Y is the corresponding instantaneous screw, the components of JTon AI, &c. A n , have intensities JT/ 7 , &c. X n ", these will generate twist velocities equal to o / c / X " &c X " g // i > ^ S ' and these quantities must equal the components of the twist velocity about Y. But the ratios are all equal, and hence the twist velocities of the com- ponents on the screws of reference of the twisting motion about Fmust be proportional to the intensities of the components on the same screws of reference of the wrench on X. Remembering that twisting motions and wrenches are compounded by the same rules, it follows that Y and X must be identical. As it is not generally true that all the screws of the complex defining the freedom possess the property enjoyed by a principal screw of inertia, it follows that the number of principal screws of inertia must be generally equal to the order of the freedom. 58. Kinetic Energy. The twisting motion of a rigid body with freedom of the n th order may be completely specified by the twist velocities of the components of the THE PRINCIPAL SCREWS OF INERTIA. 53 twisting motion on any n screws of the complex defining the freedom. If the screws of reference be a set of con- jugate screws of inertia, the expression for the kinetic energy of the body consists of n square terms. This will now be proved. If a free or constrained rigid body be at rest in a po- sition A, and if the body receive an impulsive wrench, the body will commence to twist about a screw a with a kinetic energy E a . Let us now suppose that a second impulsive wrench acts upon the body on a screw ju, and that if the body had been at rest in the position A, it would have commenced to twist about a screw /3, with a kinetic energy E$. We are now to consider how the amount of energy acquired by the second impulse is affected by the circum- stance tjiat the body is then not at rest in A, but is moving through A in consequence of the former im- pulse. The amount will in general differ from E ft , for the movement of the body may cause it to do work against the wrench on ^ during the short time that it acts, so that not only will the body thus expend some of the kinetic energy which it previously possessed, but the efficiency of the impulsive wrench on ju will be dimi- nished. Under other circumstances the motion through A might be of such a character that the impulsive wrench on [i acting for a given time would impart to the body a larger amount of kinetic energy than if the body were at rest. Between these two cases must lie the intermediate one in which the kinetic energy imparted is precisely the same as if the body had been at rest. It is obvious that this will happen if each point of the body at which the forces of the impulsive wrench are applied be moving in a direction perpendicular to the corresponding force, or more generally if the screw a about which the body is twisting be reciprocal to p. When this is the case 54 THE PRINCIPAL SCREWS OF INERTIA. a and /3 must be conjugate screws of inertia ( 54), and hence we infer the following theorem : If the kinetic energy of a body twisting about a screw a with a certain twist velocity be E ay and if the kinetic energy of the same body twisting about a screw j3 with a certain twist velocity be E ft , then when the body has a motion compounded of the two twisting movements, its kinetic energy will amount to E a + E$ provided that a and j3 are conjugate screws of inertia. Since this result may be extended to any number of conjugate screws of inertia, and since the terms E^ &c., are essentially positive, the required theorem has been proved. 59. Expression for Kinetic Energy. If a rigid body have a twisting motion about a screw a, with a twist velocity d', what is the expression of its kinetic energy in terms of the co-ordinates of a r We adopt as the unit of force that force which acting upon the unit of mass for the unit of time will give the body a velocity which would carry it over the unit of distance in the unit of time. The unit of energy is the work done by the unit force in moving over the unit dis- tance. If, therefore, a body of mass w have a movement of translation with a velocity v its kinetic energy ex- pressed in these units is ^wv z . The movement is to be decomposed into twisting motions about the screws of reference wi, &c. we, the twist velocity of the component on w m being d'a m . One constituent of the twisting motion about w m con- sists of a velocity of translation equal to ap m a m , and on this account the body has a kinetic energy equal to ^Ma*p m z a m ~. On account of the rotation around the axis with an angular velocity aa m the body has a kinetic energy equal to THE PRINCIPAL SCREWS OF INERTIA. 55 where r denotes the perpendicular from the element dM on w m . Remembering that p m is the radius of gyration this expression also reduces to J Md^p^m a m 2 , and hence the total kinetic energy of the twisting motion about w m is M&pJaJ. We see, therefore (58), that the kinetic energy due to the twisting motion about a is Ma z (/xW + &c. + /.W). The quantity inside the bracket is the square of a certain linear magnitude which is determined by the dis- tribution of the material of the body with respect to the screw a. It will facilitate the kinetic applications of the present theory to employ the symbol u a to denote this quan- tity. It is then to be understood that the kinetic energy of a body of mass M, animated by a twisting motion about the screw a with a twist velocity a is represented by 60. Twist Velocity acquired by an Impulse. A body of mass My which is only free to twist about a screw o, is acted upon for a short time e by a wrench of intensity tj" on a screw r\. It is required to find the twist velocity a which is acquired. Let the initial reaction of the constraints consist of a wrench of intensity X" on a screw X. Then the body moves as if it were free, but had been acted upon by a wrench of which the component on w m had the intensity "n"i\m + X^Xm. This component would generate a velocity of translation parallel to w m and equal to -^>(j/ / 7 + X // A m ). The twist velocity about w m produced by this component is found by dividing the velocity of translation by p m . On the other hand, since the co-ordinates of the screw 56 THE PRINCIPAL SCREWS OF INERTIA. a are fll , &c., a 6 , the twist velocity about u) m may also be represented by d'a m ( 36), whence If we multiply this equation by/ m 2 a m , add the six equa- tions found by giving m all values from i to 6, and re- member that a and X are reciprocal, we have ( 39,) whence a 7 is determined. This expression shows that the twist velocity pro- duced by an impulsive wrench on a given rigid body constrained to twist about a given screw, varies directly as the product of the virtual coefficient of the two screws and the intensity of the impulsive wrench, and inversely as the square of ^ . 6 1 . The Kinetic Energy acquired by an Impulse can be easily found by 59 ; for, from the last equation, hence the kinetic energy produced by the action of an impulsive wrench on a body constrained to twist about a given screw varies directly as the product of the square of the virtual coefficient of the two screws and the square of the intensity of the impulsive wrench, and inversely as* the square of u a . 62. Free Body. We shall now express the kinetic energy communicated by the impulsive wrench on rj to the body when perfectly free. The component on u) m of intensity rj'j\ m imparts a kinetic energy equal to THE PRINCIPAL SCREWS OF INERTIA. 57 whence the total kinetic energy is found by adding these six terms. The difference between the kinetic energy acquired when the body is perfectly free, and when the body is constrained to twist about a, is equal to The quantity inside the bracket reduces to the sum of 1 5 square terms, of which (p\airii - /2z?i) 2 is a specimen. The entire expression being therefore essentially posi- tive shows that a given impulse imparts greater energy to a quiescent body when free than to the same quiescent body when constrained to twist about a certain screw. 63. Lemma. If a group of instantaneous screws be- long to a complex of the n th order, then the correspond- ing group of impulsive screws also belong to a complex of the n ih order; for, suppose that n + i twisting motions about n + i screws neutralise, then the corresponding n + i impulsive wrenches must equilibrate, but this would not be possible unless all the impulsive screws belonged to a screw complex of the n th order. 64. Euler's Theorem. If a free or constrained rigid body receives an impulsive wrench, the body will com- mence to move with a larger kinetic energy when it is permitted to select its own instantaneous screw from the screw complex P defining the freedom, than it would have acquired, had it been arbitrarily restricted to any other screw of the complex. Let Q be the reciprocal complex of the (6 - n th ) order, and let P' be the screw complex of the n th order, con- sisting of those impulsive screws which, if the body were free, would correspond to the screws of P as instan- taneous screws. 58 THE PRINCIPAL SCREWS OF INERTIA. Let ri be any screw on which the body receives an impulsive wrench. Decompose this wrench into com- ponents on a system of six screws consisting of any n screws from P f y and any 6 - n screws from Q. The latter are neutralised by the reactions of the constraints, and may be omitted, while the former compound into one wrench on a screw belonging to P '; we may therefore replace the given wrench by a wrench on . Now, if the body were perfectly free, an impulsive wrench on must make the body twist about some screw a on P. In the present case, although the body is not perfectly free, yet it is free so far as twisting about a is concerned, and we may therefore, with reference to this particular im- pulse about , consider the body as being perfectly free. It follows from 62 that there would be a loss of energy if the body were compelled to twist about any other screw than a, which is the one it naturally chooses. This theorem is due to Euler.* 65. Co-ordinates of a Screw belonging to a Screw com- plex. It will now be necessary to make some extensions of the conceptions of screw co-ordinates. Suppose that a body have freedom of the n th order, we have shown that it is always possible to choose n screws from the screw complex expressing that freedom, such that each screw is reciprocal to all the rest. As an example we shall give the proof for the screw complex of the third order. Let B^ B^ B z be three screws of the reciprocal screw com- plex; then, if any screw A l be taken which is reciprocal to BI, BZ, .Z? 3 , any screw A% which is reciprocal to B ly BV, BZ, A ly and the screw A 3 , which is reciprocal to J5i 9 Bo, B z , A ly A 2 ; then the group A ly A 2 , A 3 possess the required property, and may be termed co-recipro- cals. * Thomson & Tait : Natural Philosophy, vol. i. p. 216. THE PRINCIPAL SCREWS OF INERTIA. 59 The co-ordinates of a scrciv belonging to a given screw complex are simplified by taking n co-reciprocal screws belonging" to the given screw complex as a portion of the six screws of reference. In this case, out of the six co ordinates ai, . . . . a 6 of a screw a, which belongs to the complex, 6-n are actually zero. Thus we are en- abled to give a more general definition of screw co- ordinates, which will apply to a screw-complex of every order from i to 6, both inclusive. If a wrench, of which the intensity is one unit on a screw a, which belongs to a certain screw complex of the n th order, be decomposed into n wrenches of intensities ai, . . . . a* on n co-reciprocal screws belonging to the same screw complex, then the n quantities a t , .... a,, are said to be the co-ordinates of the screw a. Thus the pitch of a will be represented by / t a? + . . . + / n a n 2 . The virtual coefficient of a and |3 will be 2 (/iaj3i + . . . +/ B aj3 n ) We may here remark that one screw can always be found upon a screw complex of the n th order reciprocal to n - i screws of the same complex. For, take 6 % - n screws of the reciprocal screw complex, then the required screw is reciprocal to 6 - n + n - i = 5 known screws, and is therefore determined ( 27). 66. The Reduced Wrench. A wrench which acts upon a constrained rigid body may always be replaced by a wrench on a screw belonging to the screw complex, which defines the freedom of the body. Take n screws from the screw complex of the n th order which defines the freedom, and 6 - n screws from the reciprocal complex. Decompose the given wrench into components on these six screws. The component wrenches on the reciprocal complex are neutralized by the reactions of the constraints, and may be discarded, while the remainder must compound into a wrench on the given screw complex. <6o THE PRINCIPAL SCREWS OF INERTIA. Whenever a given external wrench is replaced by an equivalent wrench upon a screw of the complex which .defines the freedom of the body, the latter may be termed, for convenience, the reduced wrench. It will be observed, that although the reduced wrench can always be determined from the given wrench, that the converse problem is indeterminate (n < 6). We may state this result in a somewhat different manner. A given wrench can always be resolved into two wrenches one on a screw of any given complex, and the other on a screw of the reciprocal screw com- plex. The former of these is what we denote by the reduced wrench. 67. Co-ordinates of Impulsive and Instantaneous Screws. Taking as screws of reference the n principal screws of inertia ( 57), we require to ascertain the rela- tion between the co-ordinates of a reduced impulsive wrench and the co-ordinates of the corresponding instan- taneous screw. If the co-ordinates of the reduced wrench are n/', . . ., rj n x/ , and those of the corresponding twisting motion are a/,. . , a,/, then, remembering the property of a principal screw of inertia ( 57), and denoting by MI, . . ., u n , the values of the magnitude u ( 59) for the principal screws of inertia, we have, from 60, whence we deduce the following theorem, which, in the particular case of n = 6, reduces to that of 53. If a quiescent rigid body, which has freedom of the n th order, commence to twist about a screw a, of which the co-ordinates, with respect to the principal screws of inertia, are ai, . . . a n and if/i, ...,/ be the pitches, and u ly . . ., u n the constants defined, in 59, of the THE PRINCIPAL SCREWS OF INERTIA. 6 1 principal screws of inertia, then the co-ordinates of the reduced impulsive wrench are proportional to " ''*/,.< ? Let T 1 denote the kinetic energy of the body of mass M when animated by a twisting motion about the screw a, with a twist velocity a. Let the twist velocities of the components on any n conjugate screws of inertia be de- noted by di', . . . d B '. (These screws will not be co-reci- procal unless in the special case where they are the principal screws of inertia.) It follows ( 58) that the kinetic energy will be the sum of the n several kinetic energies due to each component twisting motion. Hence we have ( 59) and also u* = Ufa? + . . . + w n W. Let Q! , . . . a ra and ]3i , . . , j3 ra be the co-ordinates of any two screws belonging to a screw complex of the n th order, referred to any n conjugate screws of inertia, whe- ther co-reciprocal or not, belonging to the same screw complex, then the condition that a and /3 should be con- jugate screws of inertia is To prove this, take the case of n = 4, and let A, B y C y D be the four screws of reference, and let A\, , A 6 be the co-ordinates of A with respect to the six principal screws of inertia of the body when free ( 52). The unit wrench on a is to be resolved into four wrenches of in- tensities ai, . . . , a 4 on A, By Cy D: each of these compon nents is again to be resolved into six wrenches on the 62 THE PRINCIPAL SCREWS OF INERTIA. screws of reference. The six co-ordinates of a, with re spect to the same screws, are therefore &C. We can now express the condition that a and ]3 are con jugate screws of inertia. This condition is ( 54) o. Denoting p?A? + . . . + p & A & * by uS, and observing that 2/iVl i^ t and similar expressions are zero, we deduce CHAPTER VII. THE POTENTIAL ENERGY OF A DISPLACEMENT. 68. The Potential Energy of a Displacement. Suppose a rigid body which possesses freedom of the n th order be submitted to the influence of any system of forces in- cluded within the restriction of 6. Let the symbol O define a position of the body from which the forces would be unable to disturb it. By a twist of amplitude 0' about a screw belonging to the screw complex of the n th order, which expresses the nature of the freedom, the body may be displaced from O to an adjacent position P, while the energy consumed in making the twist is denoted by V. It appears from 7 that the same amount of energy would be required, whatever be the route by which the movement is made from O to P. It follows that Kcan only depend on certain constants and on the position of P with respect to O. The most natural co-ordinates by which the position P can be specified with respect to O are the co-ordinates of the twist ( 34) by which the movement from O to P could be effected. In general these co-ordinates will be six in number ; but if n of the screws of reference be selected from the screw complex defining the freedom of the body, then ( 65) there will be only n co-ordinates required, and these may be de- noted by 0/, , 0/. The Potential V must therefore depend only upon certain constants relating to the forces and upon the n quantities 0/, . . . . , n ' ; and since these quantities are small, it follows that V must be capable of development in a series of ascending powers and products of the co-ordinates, whence we may write 64 POTENTIAL ENERGY OF A DISPLACEMENT. + terms of the second and higher orders, where H y H l9 ...,H n are constants, in so far as different displacements are concerned. In the first place, it is manifest that H= o ; because if no displacement be made, no energy is consumed. In the second place, H lf ---- , H n must also be each zero, because the position O is one of equilibrium ; and there- fore, by the principle of virtual velocities, the work done by small twists about the screws of reference must be zero, as far as the first power of small quantities is concerned. Finally, neglecting all terms above the second order, on account of their minuteness, we see that the function V, which expresses the potential energy of a small displacement from a position of equilibrium, is a homogeneous function of the second degree of the n co-ordinates, by which the dis- placement is defined: 69. The Wrench evoked by Displacement. When the body has been displaced to P, the forces no longer equi- librate, for a certain wrench has been evoked. We now propose to determine, by the aid of the function V, the co-ordinates of this wrench, or, more strictly, the co- ordinates of the equivalent reduced wrench ( 66) upon a screw of the complex, by which the freedom of the body is defined. If, in making the displacement, work has been done by the agent which moved the body, then the equilibrium of the body was stable when in the position O, so far as this displacement was concerned. Let the displacement screw be 0, and let a reduced wrench be evoked on a screw rj of the complex, while the intensities of the com- ponents on the screws of reference are i^", . . . . , ij n ". Suppose the body be displaced from P to an excessively close position P, the co-ordinates of P, with respect to POTENTIAL ENERGY OF A DISPLACEMENT. 65 O, being 0/ + S0/, . . . . n ' + S0 n ' ( 65). The potential V of the position P is it being understood that S0/, . . . , n ' are infinitely small magnitudes of a higher order than 0/, 0n'. The work done in forcing the body to move from Pto P' is V - V. This must be equal to the work done in the twists about the screws of reference whose am- plitudes are S0/, . . . . , S0 n ', by the wrenches on the screws of reference whose intensities are i^", . . . . , Tj n x/ . As the screws of reference are co-reciprocal, this work will be equal to ( 35) Since the expression just written must be equal to V - Vfor every position P in the immediate vicinity of P, we must have the coefficients of S0/, . . . , S0 B ' equal in the two expressions, whence we have n equations, of which the first is Hence, we deduce the following useful theorem : If a free or constrained rigid body be displaced from a position of equilibrium by twists of small amplitudes, 0J 7 , ---- , n 7 , about n co-reciprocal screws of reference, and if V denote the work done in producing this move- ment, then the reduced wrench has, for components on the screws of reference, wrenches of which the intensities are found by dividing twice the pitch of the corresponding reference screw into the differential coefficient of V F 66 POTENTIAL ENERGY OF A DISPLACEMENT. with respect to the corresponding amplitude, and chang- ing the sign of the quotient. It is here interesting to notice that the co-ordinates of the reduced impulsive wrench referred to the principal screws of inertia, which would give the body a kinetic energy T on the screw 6, are proportional to (67) 2p l d'0 l '' ' 2p n dti n ' | 70. Conjugate Screws of the Potential. Suppose that a twist about a screw 9 evokes a wrench on a screw 77, while a twist about a screw evokes a wrench on a screw . If 9 be reciprocal to , then must be reciprocal to r\. This will now be proved. The condition that 9 and are reciprocal is but the intensities (or amplitudes) of the components of a wrench (or twist) are proportional to the co-ordinates of the screw on which the wrench (or twist) acts, whence the last equation may be written UW +....+ p n 9 n 'Zn" = o ; but we have seen ( 69) that whence the condition that 9 and are reciprocal is 0> dV * * ^> dV * 01 -rf + + V n -= = O. 4f^ a^,' Now, as V$ is an homogeneous function of the second order of the quantities $\, . . . , n x , we may write POTENTIAL ENERGY OF A DISPLACEMENT. 67 in which AM *= AI&. Hence we obtain -^~ = 2 f Aufi + Anfa' + ---- 4 A ln $* Introducing these expressions we find, for the condition that and should be reciprocal. 0/(^n0i'+ . A ln n') +....+ 0n f (A n A f + ....+ ^nn0) = O. This may be written in the form : -4 aft V + . , A nn O n ' ft ' + -4 M (0iV + 0/00 + ....- o. But this equation \s perfectly symmetrical vntih respect to 9 and 0, and therefore we should have been led to the same result by expressing the condition that was reciprocal to j. When and possess this property, they are said to be conjugate screws of the potential, and the condition that they should be so related, expressed in terms of their co-ordinates, is obtained by omitting the accents from the last equation. If a screw be reciprocal to 77, then is a conjugate screw of the potential to 0. If we consider the screw to be given, we may regard the screw complex of the fifth order, which embraces all the [screws reciprocal to aj, as in a certain sense the locus of 0. All the screws conjugate to 0, and which, at the same time, belong to the screw complex C by which the freedom of the body is defined, must constitute in themselves a screw com- plex of the (n - i ) th order. For, besides fulfilling the 6-n conditions which define the screw complex C, they must also fulfil the condition of being reciprocal to ij ; but all F 2 68 POTENTIAL ENERGY OF A DISPLACEMENT. the screws reciprocal to 7 - n screws constitute a screw complex of the (n- i) th order ( 46). The reader will be careful to observe the distinction between two conjugate screws of inertia ( 54), and two conjugate screws of the potential. Though these pairs possess some useful analogies, yet it should be borne in mind that the former are purely intrinsic to the rigid body, inasmuch as they only depend on the distribu- tion of its material, while the latter involve extrinsic considerations, arising from the forces to which the body is submitted. 71. Principal Screws of the Potential. We are now going to prove that n screws can be found such that when the body is displaced by a twist about any one of these screws, a reduced wrench is evoked about the same screw. The screws which possess this property are called the principal screws of the potential. Let a be a principal screw of the potential, then we must have, 69 : 01 ' - and (n - i) similar equations. Introducing the value of V a> and remembering (36) that a/' = a"cti and a/ = a'ai, we have the following n equations : / " \ qifAu + -7^1 j + a 2 A lz + . . . +a n Am = o, &c., &c. A (A " \ + a z A nz + . . + a n ( Ann + p = O. \ o / From these linear equations Ql , a n can be elimi- nated, and we obtain an equation of the n th degree* in * All the roots of this equation are real. See Salmon's Higher Algebra, Art. dd. POTENTIAL ENERGY OF A DISPLACEMENT. 69 // // . The values of r substituted successively in the linear a a equations just written will determine the co-ordinates of the n principal screws of the potential. We can now show that these n screws are co-reci- procal. It is evident, in the first place, that if S be a principal screw of the potential, and if be a displace- ment screw which evokes a wrench on ?j, the principle of 70 asserts that, when is reciprocal to S, then must also rj be reciprocal to S. Now, let the n principal screws of the potential be denoted by Si, . . . , S n , and let T n be that screw of the screw complex which is recipro- cal to S ly . . . , , S n . i ( 65), then if the body be displaced by a twist about T n , the wrench evoked must be on a screw reciprocal to Si, . . . . , S n . i ; but T n is the only screw of the screw complex possessing this property; therefore a twist about T n must evoke a wrench on T n , and therefore T n must be a principal screw of the poten- tial. But there are only n principal screws of the potential, therefore T n must coincide with S n , and there- fore S n must be reciprocal to Si, . . . . S n - 1- 72. Co-ordinates of the "Wrench evoked by a Twist. The work done in giving the body a twist of small am- plitude a about a screw a, may be denoted by In fact, remembering that Q'OI = a/, . . . , and substituting these values for a/ in F( 70), we deduce the expression : f . . . + A nn a n z + 2A lz a l a 2 + 2 A 13 a!a 3 + . . . where F is a certain constant, whose dimensions are a mass divided by the square of a time, and where v a is a linear magnitude specially appropriate to each screw a, and 70 POTENTIAL ENERGY OF A DISPLACEMENT. depending upon the co-ordinates of a, and the constants in the function F( 68). The parameter z> a may be constrasted with the para- meter u a considered in 59. Each is a linear magni- tude, but the latter depends only upon the co-ordi- nates of a, and the distribution of the material of the rigid body. Both quantities may be contrasted with the pitch / a , which is also a linear magnitude, but depends solely on the screw, and is therefore purely geometrical. If a body receive a twist of small amplitude a' about one of the principal screws of the potential, then the in- tensity of the wrench evoked on the same screw is ( 69) : I * J / J 2p a aa. but we have just seen that F= Fv^a*, whence we have the following theorem : If a body which has freedom of the" n th order be displaced from a position of equilibrium by a twist about a screw a, of which the co-ordinates with respect to the principal screws of the potential are ai, . . . . , a ny then a wrench ( 66) is evoked on a screw of which the co-ordinates are proportional to ^-QI, . . . , a n , /l pn where v l9 &c., p ly &c., are the values of the quantity v, and the pitch p, for the principal screws of the potential. We can now express with great simplicity the con- dition that two screws 9 and shall be conjugate screws of the potential. For, if 9 be reciprocal to the screw whose co-ordinates are proportional to ^i 2 Vn Z fr we have : - O. POTENTIAL ENERGY OF A DISPLACEMENT. 7 1 The expression for the potential assumes the simple form va + ...+ If the function Vbe proportional to the product of the pitch of the displacement screw and the square of the amplitude, then every displacement screw will coincide with the screw about which the wrench is evoked. 73. Form of the Potential. The n principal screws of the potential form a unique group, inasmuch as they are co-reciprocal, as well as being conjugate screws of the potential. They therefore fulfil 15 -f 15 = 30 conditions, being the total number available in the selection of six screws. We are now going to show that the expression of the potential will consist of the sum of n square terms, whenever it is referred to any set of n conjugate screws of the potential. The energy consumed in giving a body a twist of amplitude & from the position of equilibrum O to a new position P, is equal to FvfW* (72), and TJ is the screw on which the wrench is evoked. Suppose that now from the position P the body receive a twist of amplitude $' about a screw 0, it would generally not be correct to assert that the energy consumed in the second twist was pro- portional to the square of its amplitude. For, during the second twist, either a portion of the energy will be consumed in doing work against the wrench on TJ, or the energy expended in the second twist will be ren- dered less, in consequence of the assistance afforded by the wrench on /. If, however, TJ be reciprocal to 0, then the quantity of energy consumed in the twist about will be unaffected by the presence of a wrench on ?j. Hence if 9 and be two conjugate screws of the potential, 72 POTENTIAL ENERGY OF A DISPLACEMENT. the energy expended in giving the body first a twist of amplitude 6' about 9, and then a twist of amplitude ^' about 0, is to be represented by Fofff* + Fv^. By taking a third screw, conjugate to both 9 and H n are all zero, then the equations as- sume a very simple form .: ft' /!!/?! -sin (r i/+i), &c. O n ' The interpretation of this result is very remarkable. We see that the co-ordinates of the body are always proportional to f uy . . . . ,/i n ; hence the body can al- ways be brought from the initial position to the position at any time by twisting it about that screw, whose co-ordinates are proportional to f lly . . . .,/!; but, as we have already pointed out, the screw thus defined HARMONIC SCREWS. 8 1 is an harmonic screw, and hence we have the follow- ing theorem : If a rigid body occupy a position of stable equili- brium under the influence of a system of forces, as restricted in 6, then n harmonic screws can be selected from the screw complex of the n th order, which defines the freedom of the body, and if the body be displaced from its position of equilibrium by a twist about a har- monic screw, and if it also receive a small initial twist velocity about the same screw, then the body will continue for ever to perform twist oscillations about that harmonic screw, and the amplitude of the twist will be always equal to the arc of a certain circular pendulum, which has an appropriate length, and was appropriately started. The integrals in their general form prove the follow- ing theorem : A rigid body is slightly displaced by a twist from a position of stable equilibrium under the influence of a system of forces, and the body receives a small initial twisting motion. The twist, and the twisting motion, may each be resolved into their components on the n harmonic screws : n circular pendulums are to be con- structed, each of which is isochronous with one of the harmonic screws. All these pendulums are to be started at the same instant as the rigid body, each with an arc, and an angular velocity equal to the initial amplitude of the twist, and the twist velocity, which has been assigned to the corresponding harmonic screw, as its share of the initial circumstances. To ascertain where the body would be at any future epoch, it will only be necessary to calculate the arcs of the n pendulums for that epoch, and then give the body twists from its position of equili- brium about the harmonic screws, whose amplitudes are equal to these arcs. The reader will observe that the solution to which G 82 HARMONIC SCREWS. we have been conducted possesses the features which we have pointed out in n, as characterising a complete discussion of a problem in the dynamics of a rigid body. 77. Remarks on Harmonic Screws. We may to a certain extent see the actual reason why the body, when once oscillating upon a harmonic screw, will never de- part therefrom. The body, when displaced from the position of equilibrium by a twist upon a harmonic screw 0, and then released, is acted upon by the wrench upon a certain screw ij, which is evoked by the twist. But the actual effect of an impulsive wrench on rj would be to make the body twist about the harmonic screw, and as the continued action of the wrench on i/ is indis- tinguishable from an infinite succession of infinitely small impulses, we can find in the influence of the forces no cause adequate to change the motion of the body from twisting about the harmonic screw 0. CHAPTER IX. THE DYNAMICS OF A RIGID BODY, WHICH HAS FREEDOM OF THE FIRST ORDER. 78. Introduction. In the present chapter we shall apply the principles developed in the preceding chapters to the examination of the Dynamics of a rigid body which has freedom of the first order. The ensuing chapters will be similarly devoted to the other orders of freedom. We shall in each chapter first ascertain what can be learned as to the kinematics of a rigid body, so far as small displacements are concerned, from merely knowing the order of the freedom which is permitted by the constraints. This will conduct us to a knowledge of the screw complex which exactly defines the freedom enjoyed by the body. We shall then be enabled to determine the reciprocal screw complex, which involves the theory of equilibrium. The next group of questions will be those which relate to the effect of an impulse upon a quiescent rigid body, free to twist about all the screws of the screw complex. Finally, we shall discuss the small oscillations of a rigid body in the vicinity of a position of stable equilibrium, under the influence of a given system offerees, the movements of the body being limited as before to the screws of the screw complex. 79. Screw Complex of the First Order. A body which has freedom of the first order can execute no movement which is not a twist about one definite screw. The position of a body so circumstanced is to be specified by a single datum, viz., the amplitude of the twist about the given screw, by which the body can be brought G 2 84 DYNAMICS OF A RIGID BODY. from a standard position to any other position which it is capable of attaining. As examples of a body which has freedom of the first order, we may refer to the case of a body free to rotate about a fixed axis, but not to slide along it, or of a body free to slide along a fixed axis, but not to rotate around it. In the former case the screw com- plex consists of one screw, whose pitch is zero ; in the latter case the screw complex consists of one screw, whose pitch is infinite. 80. The Reciprocal Screw Complex. The integer which denotes the order of a screw complex, and the ( integer which denotes the order of the reciprocal screw complex, will, in all cases, have the number six for their sum ( 46). Hence a screw complex of the first order will have as its reciprocal a screw complex of the fifth order. We shall, therefore, be obliged to discuss in the pre- sent chapter some properties of the screw complex of the fifth order, and so far to anticipate what would more naturally fall under Chapter XIII. For a screw 9 to belong to a screw complex of the fifth order, the necessary and sufficient condition is, that 6 be reciprocal to one given screw a. This con- dition is thus expressed : (fa +fo) cos O - d'sin O = o, where O is the angle, and d the perpendicular distance between the screws 6 and a. We can now show that every straight line in space, when it receives an appropriate pitch, constitutes a screw of a given screw complex of the fifth order. For the straight line and a being given, d and O are de- termined, and hence the pitch p e can be determined by the linear equation just written. Consider now a point A y and the screw a. Every straight line through A> when furnished with the proper DYNAMICS OF A RIGID BQDY. 85 pitch, will be reciprocal to a. Since the number oflines through A is doubly infinite, it follows that a singly in- finite number of screws of given pitch can be drawn through A, so as to be reciprocal to a. We shall now prove that all the screws of the same pitch which pass through A y and are reciprocal to a, lie in a plane. This we shall first show to be the case for all the screws of zero pitch,* and then we shall deduce the more general theorem. By a twist of small amplitude about a the point A is moved to an adjacent point B. To effect this movement against a force at A which is perpendicular to AJB, no work will be required; hence every line through ^4, per- pendicular to A By may be regarded as a screw of zero pitch, reciprocal to a. We must now enunciate a principle which applies to a screw complex of any order. We have already re- ferred to it with respect to the cylindroid (20). If all the screws of a screw complex be modified by the ad- dition of the same linear magnitude (positive or nega- tive) to the pitch of every screw, then the collection of screws thus modified still form a screw complex of the same order. The proof is obvious, for since the virtual co-efficient depends on the sum of the pitches, it follows that, if all the pitches of a complex be increased by a cer- tain quantity, and all the pitches of the reciprocal com- plex be diminished by the same quantity, then all the first group of screws thus modified are reciprocal to all the second group as modified. Hence, since a screw * This theorem is due to Mobius, who has shown, that, if small rotations about six axes can neutralise, and if five of the axes be given, and a point on the sixth axis, then the sixth axis is limited to a plane. (Ueber die Zusam- mensetzung unendlich kleiner Drehungen Crelle's Journal, t. xviii., pp. 189- 86 DYNAMICS OF A RIGID BODY. complex of the n th order consists of all the screws reci- procal to 6 - n screws, it follows that the modified group must still be a screw complex. We shall now apply this principle to prove that all the screws X of any given pitch k, which can be drawn through A, to be reciprocal to a, lie in a plane. Take a screw 17, of pitch p a + k, on the same line as a, then we have just shown that all the screws /*, of zero pitch, which can be drawn through the point A, so as to be reciprocal to r/, lie in a plane. Since fj, and 7j are reci- procal, the screws on the same straight lines as ^t and rj will be reciprocal, provided the sum of their pitches is the pitch of ij ; therefore, a screw X, of pitch , on the same straight line as ju, will be reciprocal to the screw a, of pitch / a ; but all the lines ju lie in a plane, therefore all the screws X lie in the same plane. Conversely, given a plane and a pitch k, a point A can be determined in that plane, such that all the screws drawn through A in the plane, and possessing the pitch k, are reciprocal to a. To each pitch k^ 2 , . . . . , will correspond a point A ly A z . . . . ; and it is worthy of re- mark, that all the points A ly A^ must lie on a right line which intersects a at right angles ; for join A ly A Zy then a screw on the line A^A^ which has for pitch either k or / 2 , must be reciprocal to a ; but this is. impossible unless A A 2 intersect a at a right angle. 81. Equilibrium. If a body which has freedom of the first order be in equilibrium, then the necessary and sufficient condition is, that the forces which act upon the body shall constitute a wrench on a screw of the screw complex of the fifth order, which is reciprocal to the screw which defines the freedom. We thus see that every straight line in space may be the residence of a screw, a wrench on which is consistent with the equilibrium of the body. DYNAMICS OF A RIGID BODY. 87 If two wrenches act upon the body, then the condi- tion of equilibrium is, that, when the two wrenches are compounded by the aid of the cylindroid, the single wrench which replaces them shall lie upon that one screw of the cylindroid, which is reciprocal to a ( 28). We can express with great facility, by the aid of screw co-ordinates, the condition that wrenches of in- tensities 0", 0", on two screws 0, 0, shall equilibrate, when applied to a body only free to twist about a. Adopting any six co-reciprocals as screws of refer- ence, and resolving each of the wrenches on 9 and

&c., we obtain the equation : lm UT& + m z U$ = o where U^ denotes the expression : -dU Solving the quadratic equation for / : m y we obtain two values of this ratio, and hence ( 89) we see that two screws belonging to the screw complex UQ = o can be found on any cylindroid (rj, ). If the relation between rj and be such, that the two roots of the equation will be equal in magni- tude, and opposite in sign, and hence we deduce the fol- lowing theorem : * Conic Sections, 3rd Edition, p. 134. DYNAMICS OF A RIGID BODY. 97 If the condition U^ = o be fulfilled, then the two screws TJ, , and the two screws on the cylindroid (TJ, ), which belong to the complex UQ = o, are parallel to the four rays of an harmonic pencil ( 89). We can now deduce a result of some importance. If we regard the screw r\ as being given, then the screw must belong to a screw complex of the fifth order and first degree, which is defined by : This complex may be constructed in the following manner : Draw any cylindroid through TJ, find on this the two screws which belong to U e = o, then a fourth screw ? can be determined by the condition that the set shall be parallel to the rays of an harmonic pencil. The same process repeated for four other cylindroids through 77, will give five screws, by which the screw complex to which belongs is determined. It will be observed that in the determination of the screw complex U^ = o, where TJ [is given, no occasion has arisen for making mention of the screws of reference to which the co-ordinates are referred. If, further, it be observed, that all the screws of the complex 6^ = are reciprocal to that one screw of which the co-ordinates are proportional to p\ \ dkji / ' ' A ^ we have the following theorem : If Ue - o denote a screw complex of the fifth order and second degree, then to every screw r\ corresponds with respect to the screw complex, a polar screw, whose co-ordi- nates are proportional to r "/. H 98 DYNAMICS OF A RIGID BODY. the relation between r\ and its polar being completely independent of the group of co-reciprocal screws, which have been chosen as the screws of reference. 92. Properties of Screws and their Polars. We add here a few properties which are, however, not demon- strated, as we shall have no occasion to make use of them. If a and /3 be two screws, and if r\ and be their polars, with respect to a screw complex of the fifth order and second degree, then, when a is reciprocal to , we shall find that j3 is reciprocal to r\. We may term a and |3 conjugate screws of the complex. If the discriminant of Ue = o vanish, there is then a "central" screw of the complex, to which the polars of all other screws are reciprocal. The equation of the screw complex will reduce to the sum of six square terms when referred to six screws of which each pair are conjugate. Six screws can be found which coincide with their polars, and these six screws are both conjugate and co- reciprocal. Six screws can be found, every pair of which are conjugate with respect to each of two given screw com- plexes of the fifth order and second degree. 93. Pitch Complex. All the screws in space of given pitch h belong to a screw complex of the fifth order and second degree, of which the equation may be written : p$? + . . . + p$ A ( 1 -rv ) pi arii J \ A driQ J where A and h are constants to be determined. We sacrifice no generality by making the pitch of n zero. We shall now write two identical equations. Of these the first expresses that the pitch of is p^ and the second expresses that the virtual co-efficient of % and TJ is p$ : h * h dR\ I h dR\ ~ Remembering that pw * + . . . +/ 6 r? 8 2 = o, and also that R is an homogeneous function of the second order, and that, therefore, by Euler's theorem* : dR dR j +....+ 7/ 6 = 2R = 2, dr\\ arje we have : * Williamson's Differential Calculus, 2nd Ed., p. 1 13. H 2 100 DYNAMICS OF A RIGID BODY. Whence we deduce Ah = -*% and since when p$ = o 4 the screw must reduce to TJ, we find A = i . We, therefore, deduce the following theorem : If rj!, .... *?6 be the co-ordinates of a screw of zero pitch, then the co-ordinates of a screw ?, of pitch /^, upon the same straight line as the screw j] are equal to the six quantities : ^_ ^ 4-A d"n\ 4/6 dris in which R = rji z + . . . . + r} + 2rjiTj 2 COS (cui(i> 2 ) + 2i7 1 rj 3 COS fth6 + . . . = I. We may remark that the co-ordinates of a screw of infinite pitch, parallel to ?, are proportional to : i dJR i dR We can also prove that -- is the cosine of the 2 tflft angle between the screw r;, and the screw of reference Wl . Let O be this angle, and let d be the shortest distance between 17 and e^. Then we have ( 35) : and as this must be true whatever may be the value of ps, it follows that : 1 dR - = cos O. 2 drji We also have the identity : J./'^.Y- -i^-o. ' DYNAMICS OF A RIGID BODY. IOI From this we see that three of the pitches of a set of six co-reciprocals must be +, and three must be -.* For, suppose that the pitches of four of the co-reciprocals had the same sign, and let ij be a screw perpendicular to the two remaining co-reciprocals, then the identity just writ- ten would reduce to the sum of four positive terms equal zero, which is absurd. 94. Screws on One Line. There is one case in which a body has freedom of the second order that demands special attention. Suppose the two given screws 0, ^i- Let

2 , w 3 , 014 to satisfy the following equations : (jt}\ fiJ*> &^3 sin (ft - ft) sin (03 - 00 sin (ft - ft)' sin (*- CU 2 3 ) sin (03 w I sin (0i -*r sm X 2 00 sin (ft -ft) l>3 sin (0 2 -3)' W4 sm (03- 00 sm (04 -00 sin (0 2-^3) whence sin (0j - ft) sin (ft - ft) = sin (fa -fa) sin (fa -fa] sin (ft - ft) sin (0 4 - ft) ~ sin (0 3 - 00 sin (0 4 - 2 ) J which proves the theorem. If, therefore, we are given three screws on the impul- sive cylindroid, and the corresponding three screws on the instantaneous cylindroid, the connexion between every other corresponding pair is geometrically] deter- mined. 100. Reaction of Constraints. Whatever the con- straints may be, their reaction produces an impulsive wrench ^ upon the body at the moment when the impulsive wrench X l acts. The two wrenches X and R compound into a third wrench Y\. If the body were free, Y l is the impulsive wrench to which the instanta- neous screw A l would correspond. Since X it X^ X z are cocylindroidal, A l9 A 2 , A 3 must be cocylindroidal, 108 DYNAMICS OF A RIGID BODY. and therefore also must be Y ly Y z , Y 3 . The nine wrenches X ly X 2y X 3y R ly R zy R z , - Y ly - Y zy - Y, must equilibrate ; but if X ly Xtj X z equilibrate, then the twist velocities about A i, A 2 , A 3 must neutralize, and therefore the wrenches about Y ly Y z , Y^ must equilibrate. Hence RI, Rty R 3 equilibrate, and are therefore cocylindroidal. Following the same line of proof used in the last section, we can show that If impulsive wrenches on any four cocylindroidal screws act upon a partially free rigid body, the four corresponding initial reactions of the constraints also constitute wrenches about four cocylindroidal screws ; and, further, the anharmonic ratios of the two groups of four screws are equal. 10 1. Principal Screws of Inertia. If a quiescent body with freedom of the second order receive impulsive wrenches on three screws X ly X 2y X z on the cylindroid which expresses the freedom, and if the corresponding instan- taneous screws on the same cylindroid be A l9 A Z) A^ then the relation between any other impulsive screw X on the cylindroid and the corresponding instantaneous screw A is completely defined by the condition that the anharmonic ratio of X, X ly X Zy X z is equal to the anhar- monic ratio of A, A ly A z A s . Now, if three rays parallel to X ly X- 2y X 3 be drawn from a point, and also three rays parallel to A ly A 2y A z , then it is well known* that the problem to determine a ray Z such that the anharmonic ratio of Z, A ly A 2 , A z is equal to that of Z, X ly X 2y X 3y admits of two solutions. There are, therefore, two screws on a cylindroid which possess the property that an impulsive wrench on one of these screws will cause the body to commence to twist about the same screw. * Chasles, passim. See alsoTownsend's Modern Geometry, vol. ii., p. 246, DYNAMICS OF A RIGID BODY. 109 We have thus arrived by a special process at the two principal screws of inertia posssesed by a body which has freedom of the second order. This is, of course, a particular case of the general theorem of 5 1 . We shall show in the next section how these screws can be determined in another manner. 102. The Ellipse of Inertia. We have seen ( 59) that a linear parameter u^ may be conceived appropriate to each screw a of a complex, so that when the body is twisting about the screw a with the unit of twist velocity, the kinetic energy is found by multiplying the mass of the body into the square of the line u^ We are now going to consider the distribution of this magnitude on u a the screws of a cylindroid. If we denote by Ui, u because the screws fi, v are parallel to the asymptotes of whence we find DYNAMICS OF A RIGID BODY. 1 2 1 By taking the tangent planes to the pitch quadric at the extremities of y y we should similarly find hence we deduce the very important result which may be thus stated : The three principal axes of the pitch quadric, when fur- nished with suitable pitches /, p& p y , constitute screws be- longing to the screw complex of the third order, and the equation of the pitch quadric has the form A* 2 + P$* + A 22 + AAA = - We can also show conversely that every screw of zero pitch, which belongs to the screw complex of the third order, must be one of the generators of the pitch quadric. For 9 must be reciprocal to all the screws of zero pitch on the reciprocal system of generators of the pitch quadric; and since two screws of zero pitch cannot be reciprocal unless they intersect either at a finite or infinite distance, it follows that must inter- sect the pitch quadric in an infinite number of points, and must therefore be entirely contained thereon. Let now S denote a screw complex of the third order, where a, /3, y are the three screws of the system on the principal axes of the pitch quadric. Diminish the pitches of all the screws of ,5* by any magnitude k. Then the quadric must be the locus of screws of zero pitch in the altered system, and therefore of pitch + k in the original system ( so). 122 DYNAMICS OF A RIGID BODY. Regarding k as a variable parameter, the equation just written represents the family of quadrics which constitute the screw complex S and the reciprocal screw complex S'. Thus all the generators of one system on each qua- dric, with pitch + k, constitute screws about which the body, with three degrees of freedom, can be twisted ; while all the generators of the other system, with pitch - k y constitute screws, wrenches about which would be neutralized by the reaction of the constraints. For the quadric to be a real surface it is plain that k must be greater than the least, and less than the greatest of the three quantities / a , p^ p y . Hence the pitches of all the real screws of the screw complex S are inter- mediate between the greatest and least of the three quantities p n p^p r 112. Screws through, a Given Point. We shall now show that three screws belonging to S, and also three screws belonging to S', can be drawn through any point 3/, y, z'. Substitute ^/, _/, 2', in the equation of the last article, and we find a cubic for k. This shows that three quadrics of the system can be drawn through each point of space. The three tangent planes at the point each contain two generators, one belonging to S, and the other to S'. It 'may be noticed that these three tangent planes intersect in a straight line. Two intersecting screws can only be reciprocal if they be at right angles, or if the sum of their pitches be zero. It is hence easy to see that, if a sphere be de- scribed around any point as centre, the three screws belonging to *$*, which pass through the point, intersect the sphere in the vertices of a spherical triangle which is the polar of the triangle similarly formed by the lines belonging to S'. We shall now show that one screw belonging to S DYNAMICS OF A RIGID BODY. 123 can be found parallel to any given direction. All the generators of the quadric are parallel to the cone (A -k}* + (p ft - k}f + (A - k] z- = o, and k can be determined so that this cone shall have one generator parallel to the given direction; the quadric can then be drawn, on which two generators will be found parallel to the given direction ; one of these belongs to S, while the other belongs to S'. It remains to be proved that each screw of S has a pitch which is proportional to the inverse square of the parallel diameter of the pitch quadric* Let r be the intercept on a generator of the cone (A - by the pitch quadric A* 2 then k = - but k is the pitch of the screw of S, which is parallel to the line r. Nine constants ( 49) are required for the determina- tion of a screw complex of the third order. This is the same number as that required for the specification of a quadric surface.f We hence infer, what is indeed other- *This theorem is connected with some purely geometrical theorems of Plucher, who has shown (Neue Geometric des Raumes, p. 130) that k^x* + k?y* + &z* + k\k=/1P1 2 +AP2 2 +/3P3 2 . If a parallelepiped be constructed, of which the three lines parallel to the reciprocal screws, drawn through the centre of the pitch quadric, are conterminous edges, and of which the line parallel to p is the diagonal, and if x,y, z be the lengths of the edges, and r the length of the diagonal, then we have ( 37) x y z ~ = PI, r = ? 2 , - = ? 3 . It follows that pp must be proportional to the inverse square of the parallel diameter of the quadric surface But p p must be proportional to the inverse square of the parallel diameter of the pitch quadric, and hence the equation last written must actually be the equation of the pitch quadric, when H is properly chosen. But the equation is obviously referred to three conjugate diame- ters, and hence three conjugate diameters of the pitch quadric are parallel to three co-reciprocal screws of the screw complex. We see from this that the sum of the reciprocals of DYNAMICS OF A RIGID BODY. 125 the pitches of three co-reciprocal screws is constant. This theorem will be subsequently generalised ( 136). 1 13. Screws of the Complex parallel to a Plane. Up to the present we have been analysing the screw complex by classifying the screws into groups of constant pitch. Some interesting features will be presented by adopting a new method of classification. We shall now divide the general system into groups of screws which are parallel to the same plane. We shall first prove that each of these groups con- stitutes a cylindroid. For suppose a screw of infinite pitch normal to the plane, 'then all the screws of the group parallel to the plane are reciprocal to this screw of infinite pitch. But they are also reciprocal to any three screws of the original reciprocal system ; they, therefore, form a screw complex of the second order ( 46) that is, they constitute a cylindroid. We shall prove this in another manner. A quadric containing a line must touch every plane passing through the line.* The number of screws of the complex which can lie in a given plane is, therefore, equal to the number of the quadrics of the complex which can be drawn to touch that plane. The quadric surface whose equation is touches the plane Px + Qy + Rz + S = o, when the fol- lowing condition is satisfied :f > - k] (A -*) + C(A - *) (A - * Salmon's Analytic Geometry of Three Dimensions, p. 74. t Salmon, loc. cit., p. 49. 126 DYNAMICS OF A RIGID BODY. whence it follows that two values of k can be found, or that two quadrics can be made to touch the plane, and that, therefore, two screws of the complex, and, of course, two reciprocal screws, lie in the plane. From this it follows that all the screws of the com- plex parallel to a plane must lie upon a cylindroid. For, take any two screws parallel to the plane, and draw a cylindroid through these screws. Now, this cylindroid will be cut by any plane parallel to the given plane in two screws, which must belong to the complex ; but this plane cannot contain any other screws ; there- fore, all the screws parallel to a given plane must lie upon the same cylindroid. 114. Determination of the Cylindroid. We now pro- pose to solve the following problem : Given a plane, determine the cylindroid which contains all the screws, selected from a screw complex of the third order, which are parallel to that plane. Draw through O the centre of the pitch quadric a plane A parallel to the given plane. We shall first show that the centre of the cylindroid required lies in A. Fig. 3- Let TI, T z (Fig. 3) be two points in which the two quadrics of constant pitch touch the plane of the paper, DYNAMICS OF A RIGID BODY. I2 7 which may be regarded as any plane parallel to A ; then P is the intersection of the pair of screws be- longing to the complex PT ly PT 2 , which lie in that plane, and P is the intersection of the pair of reciprocal screws P'R^ P'R* belonging to the reciprocal complex. Since P'R^ is to be reciprocal to PT^ it is essential that Ri be a right angle, similarly R 2 is a right angle. The reciprocal cylindroid, whose axis passes through P' y will be identically the same as the cylindroid belonging to the complex whose axis passes through P; but the two will be differently posited. If the angle at P be a right angle, the points 7i and T 2 are at infinity; therefore, the plane touches the quadric at infinity; it must, therefore, touch the asymptotic cone, and must, therefore, pass through the centre of the pitch quadric O; but P is the centre of the cylindroid in this case, and, therefore, the centre of the cylindroid must lie in the plane A . The position of the centre of the cylindroid in the plane A is to be found by the following construction : Draw through the centre O a diameter conjugate in the pitch quadric to the plane A. Let this line intersect the pitch quadric in the points P ly P 2 , and let S, S' (Fig. 4) be the feet of the per- T pendiculars let fall from P iy P 2 upon the plane A . Draw the asymptotes OL, OM to the section of the pitch quad- ric, made by the plane A . Through *$* and S f draw lines in the plane A, ST, ST' y S'T y S'F, parallel to the asymptotes, then T' and T are the two required cylindroids which belong to the two reci- procal screw complexes. This construction is thus demonstrated : Fig. 4. centres of the 1 28 DYNAMICS OF A RIGID BODY, The tangent planes at P ly P* each intersect the sur- face in lines parallel to OL, OM. Let us call these lines PII, 1 ly M l through the point P ly and P Z L Z , P 2 M 2 through the point jP 2 . Then PiL ly P z M 2y are screws belonging to the complex, and PiM ly P^L Z are reciprocal screws. Since OL is a tangent to the pitch quadric, it there- fore must be intersected by two rectilinear generators,, one of each system. These two generators lie in a plane which contains OL ; but since OL touches the hyperboloid at infinity, the lines on the surface must be parallel to OL, and therefore their projections on the plane of A must be S' T, S'T'. Similarly for ST, S'T'; hence ST' and S'T' are the projections of two screws belonging to the complex, and therefore the centre of the cylindroid is at T '. In a similar way it is proved that the centre of the reciprocal cylindroid is at T. Having thus determined the centre of the cylindroid, the remainder of the construction is easy. The pitches of two screws on the surface must be proportional to the inverse square of the parallel diameters of the ^sec- tion of the pitch quadric made by A. Therefore, the greatest and least pitches will be on screws parallel to the principal axes of the section. Hence, lines drawn through T' parallel to the external and internal bisectors of the angle between the asymptotes are the two rectan- gular screws of the cylindroid. Thus the problem of finding the cylindroid is completely solved. It is easily seen that each cylindroid touches each of the quadrics in two points. 115. Miscellaneous Remarks. It follows from the last article that any plane which contains a pair of screws belonging to the complex which intersect at right angles must pass through the centre of the pitch quadric. We are now in a position to determine the actual DYNAMICS OF A RIGID BODY. 129 situation of a screw 6 belonging to a screw complex of the third order of which the direction is given. The con- struction is as follows : Draw through O the centre of the pitch quadric a radius vector OR parallel to the given direction of 9, and cutting the pitch quadric in R. Draw a tangent plane to the pitch quadric in R. Then the plane A through OR, of which the intersection with the tangent plane is perpendicular to OR, is the plane which contains 6. For the section in which A cuts the pitch quadric has for a tangent at R a line perpendicu- lar to OR; hence the line OR is a principal axis of the section, and hence ( 114) one of the two screws of the complex in the plane A must be parallel to OR. It remains to find the actual situation of in the plane A . Since the direction of is known, its pitch is deter- minate, because it is inversely proportional to the square of OR. Hence the quadric can be constructed, which is the locus of all the screws which have the same pitch as 0. This quadric must be intersected by the plane A in two parallel lines. One of these lines is the required resi- dence of the screw 0, while the other line, with a pitch equal in magnitude to that of 0, but opposite in sign, belonging, as it does, to one of the other system of generators, is a screw reciprocal to the system. A family of quadric surfaces of constant pitch have the same planes of circular section, and therefore every plane through the centre cuts the quadrics in a system of conies having the same directions of axes. The cylindroid which contains all the screws of the screw complex parallel to one of the planes of circu- lar section must be composed of screws of equal pitch. A cylindroid in this case reduces to a plane pencil of rays passing through a point. We thus have two points situated upon the primary axis of the pitch quadric,. K 130 DYNAMICS OF A RIGID BODY. through each of which a plane pencil of screws can be drawn, which belong to the screw complex. All the screws passing through either of these points have equal pitch. The pitches of the two pencils are equal in mag- nitude, but opposite in sign. The magnitude is that of the pitch of the screw situated on the primary axis of the pitch quadric.* 1 1 6. Virtual Co-efficients. Let p be a screw of the screw complex which makes angles whose cosines are f, g, h, with the three screws of reference a, /3, y upon the axes of the pitch quadric. Then, reference being made to any six co-reciprocals, we have for the co- ordinates of /o, &c., &c., Let j be any given screw. The virtual co-efficient of and i is Draw from the centre of the pitch quadric a radius vec- tor r parallel to /o, and equal to the virtual coefficient just written ; then the locus of the extremity of r is the sphere = 2 The tangent plane to the sphere obtained by equating the right-hand side of this equation to zero is the prin- * If a, 5, c be the three semiaxes of the pitch quadric, and + d the distances from the centre, on R, x. A cone of screws can be drawn through eevry point K 2 132 DYNAMICS OF A RIGID BODY. in space belonging to this complex, and on that cone one screw of zero pitch can always be found. Hence one line can be drawn through every point in space along which S might act. If the body have freedom of the fourth order, the lati- tude in the choice of S is still greater. Let X ly X z be two screws reciprocal to the complex, then S is only restrained by the condition that it belong to the screw complex of the fifth order defined by the screws P, Q, X, X 19 X,. Any line in space when it receives the proper pitch is a screw of this complex. Through any point in space a plane can be drawn such that every line in the plane passing through the point with zero pitch is a screw of the complex ( 80). Finally, if the body has only freedom of the third order, the four equilibrating forces P, Q, R, S may be si- tuated anywhere. The positions of the forces being given, their magni- tudes are determined ; for draw three screws X ly X z , X$ reciprocal to the complex, and find ( 30) the intensities of the seven equilibrating wrenches on The last three are neutralised by the reactions of the constraints, and the four former must therefore equili- brate. Given any four screws in space, it is possible for four wrenches of proper intensities on these screws to hold a body having freedom of the third order in equilibrium. For, take the four given screws, and three reciprocal screws. Wrenches of proper intensities on these seven screws will equilibrate ; but those on the reciprocal screws DYNAMICS OF A RIGID BODY. 133 are destroyed by the reactions, and, therefore, the four wrenches on the four screws equilibrate. It is mani- fest that this theorem may be generalised into the fol- lowing: If a body have freedom of the k th order, then properly selected wrenches about any k + i screws (not reciprocal to the screw complex) will hold the body in equilibrium. That a rigid body with freedom of the third order may be in equilibrium under the action of gravity, we have the necessary and sufficient condition, which is thus stated : The vertical through the centre of inertia must be one of the reciprocal system of generators on the pitch quadric. We see that the centre of inertia must, therefore, lie upon a screw of zero pitch which belongs to the screw complex ; whence we have the following theorem : The restraints which are necessary for the equilibrium of a body which has freedom of the third order under the action of gravity, would permit rotation of the body round one definite line through the centre of inertia. 119. The Ellipsoid of Inertia. The momental ellip- soid, which is of such significance in the theory of the rotation of a rigid body about a fixed point, is presented in the Theory of Screws as a particular case of another ellipsoid called the ellipsoid of inertia, which is of great importance in connexion with the general screw com- plex of the third order. If we take three conjugate screws of inertia from the screw complex, as screws of reference, then we have seen (67) that, if ft, ft, ft, be the co-ordinates of a screw $, we have where u ly u^ u 3 are the values of u d with reference to the three conjugate screws of inertia. 134 DYNAMICS OF A RIGID BODY. Draw from any point lines parallel to 0, and to- the three conjugate screws of inertia. If then a pa- rallelepiped be constructed of which the diagonal is the line parallel to 0, and of which the three lines parallel to the conjugate screws are contermi- nous edges, and if r be the length of the diagonal, and x y y, z the lengths of the edges, then we have x fi y fi z a r =i 'r = ' r = 3 * We see, therefore! that the parameter u appropriate to any screw 9 is inversely proportional to the parallel diameter of the ellipsoid where His & certain constant. Hence we have the following theorem : The kinetic energy of a rigid body, when twisting with a given twist velocity about any screw of a complex of the third order, is proportional to the inverse square of the parallel dia- meter of a certain ellipsoid, which may be called the ellipsoid of inertia ; and a set of three conjugate diame- ters of the ellipsoid are parallel to a set of three conjugate screws of inertia which belong to the screw complex. We might also enunciate the property in the follow- ing manner : Any diameter of the ellipsoid of inertia is proportional to the twist velocity with which the body should twist about the parallel screw of the screw com- plex, so that its kinetic energy shall be constant. 1 20. The Principal Screws of Inertia. It will simplify matters to consider that the ellipsoid of inertia is con- centric with the pitch quadric. It will then be possible to find a triad of common conjugate diameters to the two ellipsoids. We can then determine three screws of the complex parallel to these diameters ( H5)> DYNAMICS OF A RIGID BODY. 135 and these three screws will be co-reciprocal, and also conjugate screws of inertia. They will, therefore ( 57), form what we have termed the principal screws of inertia. When the screw complex reduces to a pencil of screws of zero pitch passing through a point, then the principal screws of inertia reduce to the well-known principal axes. 121. Lemma. If from a screw complex of the n ih order we select n screws A ly . . . , A n , which are conju- gate screws of inertia ( 57), and if Si be any screw which is reciprocal to A 2 , ..., A ny then an impulsive wrench on Si will cause the body, when only free to twist about the screws of the complex, to commence to twist about A i. Let RI be the screw which, if the body were perfectly free, would be the impulsive screw correspond- ing to A i as the instantaneous screw. jRi must be reci- procal to A 2 , . . . , A n ( 54). Take also 6 - n screws of the reciprocal system B l9 . . . , B 6 _ . Then the 8 - n screws R i9 S lf ly . . . , 2? 6 _ w must be reciprocal to the n - i screws A z , . . . A n , and therefore the 8 - n screws must belong to a screw complex of the (7 - n) th order. Hence an impulsive wrench upon the screw ^i can be resolved into components on R l9 Bi 9 . . . B* _ . Of these all but the first are neutralised by the reactions of the constraints, and by hypothesis the effect of an im- pulsive wrench upon R is to make the body commence to twist about AI, and therefore an impulsive wrench on Si would make the body twist about A\. 122. Relation between the Impulsive Screw and the In- stantaneous Screw. A quiescent rigid body which pos- sesses freedom of the third order is acted upon by an impulsive wrench about a given screw r\. It is required to determine the instantaneous screw 0, about which the body will commence to twist. The screws which belong to the complex, and are at 136 DYNAMICS OF A RIGID BODY. the same time reciprocal to TJ, must all lie upon a cylin- droid, as they each fulfil the condition of being recipro- cal to four screws. All the screws on the cylindroid are parallel to a certain plane drawn through the centre of the pitch quadric, which may be termed the reciprocal plane with respect to the screw TJ. The reciprocal plane having been found, the diameter conjugate to this plane in the ellipsoid of inertia is parallel to the required screw 6. For let JJL and v denote two screws of the complex parallel to a pair of conjugate diameters of the ellipsoid of inertia in the reciprocal plane. Then 0, p, v are a triad of conjugate screws of inertia ; but r? is reciprocal to fj. and v, and, therefore, by the lemma of the last article, an impulsive wrench upon r\ will make the body commence to twist about 0. 123. Kinetic acqEnergy uired by an Impulse. We shall now consider the following problem : A quies- cent rigid body of mass M receives an impulsive wrench of intensity rj" on a screw TJ for a short time e. De- termine the locus of a screw 8 belonging to a screw complex of the third order, such that, if the body be con- strained to twist about 0, it shall acquire a given kinetic energy K, in consequence of the impulsive wrench. We have from 6 1 the equation & ri" 2 K = ^r -V M uf We can assign a geometrical interpretation to this equation, which will lead to some interesting results. Through the centre O of the pitch quadric the plane A reciprocal to t\ is to be drawn. A sphere ( 1 1 6) is to be described touching the plane A at the origin O, the diameter of the sphere being so chosen that the intercept OP made by the sphere on a radius vector DYNAMICS OF A RIGID BODY. 137 parallel to any screw is equal to ^9 ( 1 16). The quan- tity u is inversely proportional to the radius vector OQ of the ellipsoid of inertia, which is parallel to ( 119). Hence for all the screws of the screw complex which acquire a given kinetic energy in consequence of a given impulse, we must have the product OP. OQ con- stant. From a well-known property of the sphere, it follows that all the points Q must lie upon a plane A', parallel to A. This plane cuts the ellipsoid of inertia in an ellipse, and all the screws required must be parallel to the generators of the cone of the second degree, formed by joining the points of this ellipse to the origin, O. Since we have already shown how, when the direc- tion of a screw belonging to a screw complex of the third order is given, the actual situation of that screw is determined ( 1 1 5), we are now enabled to ascertain all the screws on which the body acted upon by a given impulse would acquire a given kinetic energy. The distance between the planes A and A' is pro- portional to OP. OQ, and therefore to the square root of K. Hence, when the impulse is given, the kinetic energy acquired on a screw determined by this construction is greatest when A and A' are as remote as possible. For this to happen, it is obvious that A' will just touch the ellipsoid of inertia. The group of screws will, there- fore, degenerate to the single screw parallel to the dia- meter of the ellipsoid of inertia conjugate to A. But we have seen ( 122) that the screw so determined is the screw which the body will naturally select if permitted to make a choice from all the screws of the complex of the third order. We thus see again what Euler's theorem ( 64) would have also told us, viz., that when a quies- cent rigid body which has freedom of the third order is 138 DYNAMICS OF A RIGID BODY. set in motion by the action of an impulsive wrench, the kinetic energy which the body acquires is greater than it would have been had the body been restricted to any other screw of the complex than that one which it natu- rally chooses. 124. Reaction of the Constraints. An impulsive wrench on a screw r; acts upon a body with freedom of the third order, and the body commences to move by twisting upon a screw 0. It is required to find the screw A, a wrench on which constitutes the initial reaction of the constraints. Let ^ denote the impulsive screw which, if the body were free, would correspond to 9 as the in- stantaneous screw. Then A. must lie upon the cylin- droid (0, ij), and may be determined by choosing on (0, r?) a screw reciprocal to any screw of the given screw complex. 125. Impulsive Screw is Indeterminate. Being given the instantaneous screw 6 in a complex of the third order, the corresponding impulsive screw rj is indeter- minate, because the impulsive wrench may be com- pounded with any reactions of the constraints. In fact TJ may be any screw selected from a screw complex of the fourth order, which is thus found. Draw the diame- tral plane conjugate to a line parallel to 9 in the ellipsoid of inertia, and construct the cylindroid which consists of screws belonging to the screw complex parallel to this diametral plane. Then any screw which is reci- procal to this cylindroid will be an impulsive screw cor- responding to 9 as an instantaneous screw. Thus we see that through any point in space a whole cone of screws can be drawn, an impulsive wrench on any one of which would make the body commence to twist about the same screw. One impulsive couple can always be found which would make the body commence to twist about any DYNAMICS OF A RIGID BODY. 139 given screw of the screw complex. For a couple in a plane perpendicular to the nodal line of a cylindroid may be regarded as a wrench upon a screw recipro- cal to the cylindroid ; and hence a couple in a diame- tral plane of the ellipsoid of inertia, conjugate to the diameter parallel to the screw 0, will make the body commence to twist about the screw 9. It is somewhat remarkable that a force directed along the nodal line of the cylindroid must make the body commence to twist about precisely the same screw as the couple in a plane perpendicular to the nodal line. If a cylindroid be drawn through two of the principal screws of inertia, then an impulsive wrench on any screw of this cylindroid will make the body commence to twist about a screw on the same cylindroid. For the impul- sive wrench may be resolved into wrenches on the two principal screws. Each of these will produce a twisting motion about the same screw, which will, of course, compound into a twisting motion about a screw on the same cylindroid. 126. Ellipsoid of ihe Potential. A body which has freedom of the third order is in equilibrium under the influence of a system of forces in conformity with the restrictions of 6. The body receives a twist of small amplitude & about a screw 9 of the screw complex. It is required to determine a geometrical representation for the quantity of work which has been done in effect- ing the displacement. We have seen that to each screw 9 corresponds a certain linear parameter v 9 ( 72), and that the work done is represented by We have also seen that the quantity VQ- may be repre- sented by 140 DYNAMICS OF A RIGID BODY. where ft, ft, ft are the co-ordinates of the screw referred to three conjugate screws of the potential, and v ly v 2y v 3 , denote the values of ve for each of the three screws of reference ( 72). Drawing through the centre of the pitch quadric three axes parallel to the three screws of reference, we can then construct the ellipsoid of which the equation is which proves the following theorem. The work done in giving the body a twist of given amplitude from a position of equilibrium about any screw of a complex of the third order, is proportional to the inverse square of the parallel diameter of a certain ellipsoid which we may call the ellipsoid of the potential, and three conjugate diameters of this ellipsoid are paral- lel to three conjugate screws of the potential in the screw complex. 127. The Principal Screws of the Potential. The three common conjugate diameters of the pitch hyperboloid, and the ellipsoid of the potential, are parallel to three screws of the complex which are what we call the prin- cipal screws of the potential. If the body be displaced by a twist about a principal screw of the potential from a position of stable equilibrium, then the reduced wrench which is evoked is upon the same screw. The three principal screws of the potential must not be confounded with the three screws of the complex which are parallel to the principal axes of the ellipsoid of the potential. The latter are the screws on which a twist of given amplitude requires a maximum or mini- mum consumption of energy, and they are rectangular, which, of course, is not in general the case with the principal screws of the potential. 128. Wrench evoked by Displacement. By the aid of DYNAMICS OF A RIGID BODY. 141 the ellipsoid of the potential we shall be able to solve the problem of the determination of the screw on which a wrench is evoked by a twist about a given screw 6 of the complex. The construction which will now be given will enable us to determine the screw of the complex on which the reduced wrench acts. Draw through the centre of the pitch quadric a line parallel to 9. Construct the diametral plane A of the ellipsoid of the potential conjugate to this line, and let X, fj. be any two screws of the complex parallel to a pair of conjugate diameters of the ellipsoid of the poten- tial which lie in the plane A. Then the required screw is parallel to that diameter of the pitch quadric which is conjugate to the plane A. For will then be reciprocal to both X and /* ; and as X, //, 9 are conjugate screws of the potential, it fol- lows that a twist about 9 must evoke a reduced wrench on $. 129. Harmonic Screws. When a rigid body has free dom of the third order, it must have ( 74) three harmonic screws, or screws which are conjugate screws of inertia L as well as conjugate screws of the potential. We are now enabled to construct these screws with facility, for they must be those screws of the screw complex which are parallel to the triad of common conjugate diameters of the ellipsoid of inertia, and the ellipsoid of the poten- tial. We have thus a complete geometrical conception of the small oscillations of a rigid body which has free- dom of the third order. If the body be once set twisting about one of the harmonic screws, it will continue to twist thereon for ever, and in general its motion will be compounded of twisting motions upon the three har- monic screws. If the displacement of the body from its position of 142 DYNAMICS OF A RIGID BODY. equilibrium has been effected by a small twist about a screw on the cylindroid which contains two of the har- monic screws, then the twist can be decomposed into components on the harmonic screws, and the instanta- neous screw about which the body is twisting at any epoch will oscillate backwards and forwards upon the cylindroid, from which it will never depart. If the periods of the twist oscillations on two of the harmonic screws coincided, then every screw on the cylindroid which contains those harmonic screws would also be a harmonic screw. If the periods of the three harmonic screws were equal, then every screw of the complex would be a har- monic screw. 130. Oscillations of a Rigid Body about a Fixed Point. We shall conclude the present Chapter by applying the principles which it contains to the development of a geometrical solution of the following important problem : A rigid body, free to rotate in every direction around a fixed point y is at rest under the influence of gravity. The body is slightly disturbed : it is required to determine the nature of its small oscillations. Since three co-ordinates are required to specify the position of a body when rotating about a point, it fol- lows that the body has freedom of the third order. The screw complex, however, assumes a very extreme type, for the pitch quadric has become illusory, and the screw complex reduces to a pencil of screws of zero pitch radiating in all directions from the fixed point. The quantity U Q appropriate to a screw reduces to the radius of gyration when the pitch of the screw is zero ; hence the ellipsoid of inertia reduces in the pre- sent case to the well-known momental ellipsoid. The ellipsoid of the potential ( 126) assumes a DYNAMICS OF A RIGID BODY. 143 remarkable form in the present case. The work done in giving the body a small twist is proportional to the vertical distance through which the centre of inertia is elevated. Now, as in the position of equilibrium the centre of inertia is vertically beneath the point of sus- pension, it is obvious from symmetry that the ellipsoid of the potential must be a surface of revolution about a vertical axis. It is further evident that the vertical radius vector of the ellipsoid must be infinite, because no work is done in rotating the body around a vertical axis. Let O be the centre of suspension, and / the cen- tre of inertia, and let OP be a radius vector o the ellipsoid of the potential. Let fall IQ perpendicular on OP y and PT perpendicular upon OI. It is extremely easy to show that the vertical height through which / is raised is proportional to /(> x OP* ; whence the area of the triangle OPI is constant, and therefore the locus of P must be a right circular cylinder of which Olis the axis. We have now to find the triad of common conjugate diameters of the momental ellipsoid, and the circular cylinder just described. A group of three conjugate dia- meters of the cylinder must consist of the vertical axis, and any two other lines through the origin, which are conjugate diameters of the ellipse in which their plane cuts the cylinder. It follows that the triad required will consist of the vertical axis, and of the pair of common conjugate diameters of the two ellipses in which the plane conjugate to the vertical axis in the momental ellipsoid cuts the momental ellipsoid and the cylinder. These three lines are the three harmonic axes. With reference to the vertical axis which appears to be one of the harmonic axes, the time of vibration would 144 DYNAMICS OF A RIGID BODY. be infinite, so we reject it. The three harmonic screws which are usually found in the small oscillations of a body with freedom of the third order are therefore reduced in the present case to two, and we have the following' theorem : A rigid body which is free to rotate about a fixed point is at rest under the action of gravity. If a plane S be drawn through the point of suspension O, con- gate to the vertical diameter OI of the momental ellip- soid, then the common conjugate diameters of the two ellipses in which S cuts the momental ellipsoid, and a circular cylinder whose axis is OI, are the two har- monic axes. If the body be displaced by a small rota- tion about one of these axes, the body will continue for ever to oscillate to and fro upon this axis, just as if the body had been actually constrained to move about this axis. To complete the solution for any initial circum- stances of the rigid body, a few additional remarks are necessary. Assuming the body in any given position of equili- brium, it is first to be displaced by a small rotation about an axis OX. Draw the plane containing OI and OX, and let it cut the plane S in the line OY. The small rotation around OX may be produced by a small rota- tion about OI, followed by a small rotation about OY. The effect of the small rotation about OI is merely to alter the azimuth of the position, but not to disturb the equili- brium. Had we chosen this altered position as that position of equilibrium from which we started, the ini- tial displacement will be communicated by a rotation around OY. We may, therefore, without any sacrifice of generality, assume that the axis about which the initial displacement is imparted lies in the plane *$*. We shall now suppose the body to receive a small angular DYNAMICS OF A RIGID BODY. 145 velocity about any other axis. This axis must be in the plane S, if small oscillations are to exist at all, for the initial angular velocity, if not capable of being resolved completely on the two harmonic axes, will have component around the vertical axis OL The effect of an initial rotation about OI will be to give the body a continuous slow rotation around the vertical axis, which is, of course, not admissible when small oscillations only are considered. If, therefore, the body performs small oscillations only, we may regard the initial axis of displacement as lying in the plane S, while we must have the initial instan- taneous axis in that plane. The initial displacement may be resolved into two displacements, one on each of the harmonic axes, and the initial angular velocity may also be resolved into two angular velocities on the two harmonic axes. The entire motion will, therefore, be found by compounding the vibrations about the two harmonic axes. Also the instantaneous axis will at every instant be found in the plane of the harmonic axes, and will oscillate to and fro in their plane. Since conjugate diameters of an ellipse are always projected into conjugate diameters of the projected ellipse, it follows that the harmonic axes must pro- ject into two conjugate diameters of a circle on any horizontal plane. Hence we see that two vertical planes, each containing one of the harmonic axes, are at right angles to each other. We have thus obtained a complete solution of the problem of the small oscillations of a body about a fixed point under the influence of gravity. ( 146 CHAPTER XII. THE DYNAMICS OF A RIGID BODY WHICH HAS FREEDOM OF THE FOURTH ORDER. 131. Screw Complex of the Fourth Order. The most general type of a screw complex of the fourth order is merely a group of screws which are reciprocal to an arbitrary cylindroid ( 49). To obtain a clear idea of this screw complex it is, therefore, only required to re-state a few results already obtained. All the screws belonging to a screw complex of the fourth order which can be drawn through a given point lie on a cone of the second degree ( 25). All the screws of given pitch belonging to a screw complex of the fourth order must intersect two fixed lines, viz., the two screws on the reciprocal cylindroid of pitch equal in magnitude, and opposite in sign, to the given pitch ( 24). One screw of given pitch belonging to a screw com- plex of the fourth order can be drawn through each point in space ( 97). 132. Screws Parallel to a Given Line. It is required to determine the locus of the screws parallel to a given straight line Z, which belong to a screw complex of the fourth order. This easily appears from the principle that each screw of the screw complex must intersect one screw of the reciprocal cylindroid at right angles ( 24). Take, therefore, that one screw on the cylindroid which is perpendicular to L. Then a plane through parallel to L is the required locus. 133. Screws in a Plane. As we have already seen that two screws belonging to a screw complex of the third order can be found in any plane ( 113), so we might DYNAMICS OF A RIGID BODY. 147 expect to find that a singly infinite number of screws belonging to a screw complex of the fourth order can be found in any plane. We shall now prove that all these -screws envelope a parabola. Take any point P in the plane, then the screws through P reciprocal to the cylindroid form a cone of the second order, which is cut by the plane in two lines. Thus two screws belonging to a given screw complex of the fourth order can be drawn in a given plane through a given point. From the last article it follows that only one screw of the complex parallel to a given line can be found in the plane. Therefore, the envelope must be a parabola. 134. Property of the Pitches of Six Co-reciprocals. We may here introduce an important property of the pitches of a set of co-reciprocal screws selected from a .screw complex. There is one screw on a cylindroid of which the pitch is a maximum, and another screw of which the pitch is a minimum. These screws are parallel to the principal axes of the pitch conic ( 20). Belonging to a screw complex of the third order we have, in like manner, three screws of maximum or miminum pitch, which lie along the three principal axes of the pitch quadric ( 1 1 1). The general question, therefore, arises, as to whether it is always possible to select from a screw complex of the n th order a certain number of screws of maximum or minimum pitch. Let 0j, . . . . O n be the n co-ordinates of a screw re ferred to n co-reciprocal screws belonging to the given screw complex. Then the function p^ or is to be a maximum, while, at the same time, the co-ordi- nates satisfy the condition ( 37) L 2 1 48 DYNAMICS OF A RIGID BODY. 2ft 2 + 2S0ift COS (on, Wz) = I, which for brevity we denote as heretofore by I? -i, Applying the ordinary rules* for maxima and minima, we deduce the n equations &c. &c., From these ?z linear equations it would seem that ft, . . ., can be eliminated, and that an algebraic equa- tion of the n th degree would remain for p e . The analysis would, therefore, appear to have proved that n screws of maximum or minimum pitch can always be selected from a screw complex of the n th order. A moment's reflection will, however, show that this statement needs modification. Take the case of n = 6 : the screw complex of the sixth order is simply another name for every screw in space. In this case, therefore, all the values of p e must be infinite, which implies that each co-efficient of the equation for p 9 must vanish, except the absolute term. We are thus presented with no fewer than six for- mulae involving the pitches and angles of inclination of the six screws of a co-reciprocal system. Of these for- mulae we shall in this place only consider one. If the co-efficient oip e be equated to zero it appears that i i i i i i h i + i + = o Williamson's Differential Calculus, 2nd Edition, p. 189. DYNAMICS OF A RIGID BODY. 149 or, the sum of the reciprocals of the pitches of the six screws of a co-reciprocal system is equal to zero. 135. Another Proof. The following elegant proof of the theorem of the last section was communicated to me by my friend Professor Everett. Divide} the six co-reciprocals into any two groups A and B of three each, then it appears from 1 1 1 that the pitch qua- dric of each of these groups is identical. The three screws of A are parallel to a triad of conjugate diameters of the pitch quadric, and the sum of the reciprocals of the pitches is proportional to the sum of the squares of the conjugate diameters ( 112). The three screws of B are parallel to another triad of conjugate diameters of the pitch quadric, and the sum of the reciprocals of the pitches, with their signs changed, is proportional to the sum of the squares of the conjugate diameters. Remem- bering that the sum of the squares of the two sets of conjugate diameters is equal, the required theorem is at once evident. 136. Property of the Pitches of n Co-reciprocals. The theorem just proved can be extended to show that the sum of the reciprocals of the pitches of n co-reciprocal screws, selected from a screw complex of the n th order, is a constant for each screw complex. Let A be the given screw complex, and B the reci- procal screw complex. Take 6 - n co-reciprocal screws on B, and any n co-reciprocal screw on A . The sum of the reciprocals of the pitches of these six screws must be always zero ; but the screws on B may be constant, while those on A are changed, whence the sum of the reciprocals of the pitches of the n co-reciprocal screws on A must be constant. Thus, as we have already seen from geometrical con- siderations, that the sum of the reciprocals of the pitches of co-reciprocals is constant for the screw complex of 150 DYNAMICS OF A RIGID BODY. the second and third order ( 42, 112), so now we see- that the same must be likewise true for the fourth, fifth,, and sixth orders. The actual value of this constant for any given screw- complex is evidently a characteristic feature of that screw complex. 137. Special Screw of the Complex. In general there- is one line in each csrew complex of the fourth order, which forms a screw belonging to the screw complex, whatever be the pitch assigned to it. The line in ques- tion is the nodal line of the reciprocal cylindroid. The kinematical statement is as follows : When a rigid body has freedom of the fourth order, there is in general one straight line, about which the body can be rotated, and parallel to which it can be translated. 138. Particular Case. A body which has freedom of the fourth order may be illustrated by the case of a rigid body, one point P of which is constrained to a certain curve. The position of the body will then be specified by four quantities, viz., the arc of the curve from a fixed origin up to P, and three rotations about three axes intersecting in P. The reciprocal cylindroid will in this case assume an extreme form; it consists of screws of zero pitch on all the normals to the curve at P. 139. Statics. When a rigid body has freedom of the fourth order, the necessary and sufficient condition for equilibrium is, that the forces shall constitute a wrench upon a screw of the cylindroid reciprocal to the given screw complex. Thus, if one force can act on the body without disturbing equilibrium, then this force must lie on one of the two screws of zero pitch on the cylindroid. If there were no real screws of zero pitch on the cylindroid that is, if the pitch conic were an ellipse, then it is impossible for equilibrium to subsist when a force acts. It is, however, worthy of remark,. DYNAMICS OF A RIGID BODY. 1 5 i that if one force can act without disturbing the equili- brium, then another force (on the other screw of zero pitch) will be in the same predicament. A couple which is in a plane perpendicular to the nodal line can be neutralized by the reaction of the constraints, and is, therefore, consistent with equili- brium. In no other case, however, can a body which has freedom of the fourth order be in equilibrium under the influence of a couple. 140. Equilibrium of Five Forces. The five forces must, if the body be free, belong to a screw complex of the fourth order. Draw the cylindroid reciprocal to the com- plex. The five forces must, therefore, intersect both the screws of zero pitch on the cylindroid. We, therefore, have the well-known condition that two straight lines can be drawn which intersect all the five forces. Four of the forces will determine the two lines, and therefore the fifth force may enjoy any liberty consistent with the requirement that it also intersects the two lines. This condition is also a sufficient one, so far as the positions of the forces are concerned. If A j, ... A 5 be the five forces, the ratio of A l : A* is thus determined. Let P, Q be the two screws of zero pitch upon the cylindroid. Let X, Ybe two screws reciprocal to'^, A z . Let Z be a screw reciprocal to A 3 , A^ A 5 . Construct the screw / reciprocal to the five screws X, Y, P, Q, Z. Now, the four screws X, Y, P, Q are reciprocal to the cylindroid A l9 A z ; therefore 7, which is reciprocal to X, Y y R y P y Q, must lie upon the cylindroid (Ai,A t ). Since P, Q, Z are all reciprocal to A z , A^ A s , it fol- lows that / being reciprocal to P, Q, Z must belong to 152 DYNAMICS OF A RIGID BODY. the screw complex As, A if A 5 . Hence / belongs to (A i, A z ), and also to (A 3 , A^ A 6 ). If, therefore, forces along A ly ... A 5 equilibrate, then the forces along A i, A 2 must compound into a wrench on /. This condi- tion determines the forces on A i9 A 2 ( 17). 141. Problem. A free rigid body is acted upon by five forces : show how to move the body so that it shall not do work against nor receive energy from any one of the forces. Let A i, . . . A 5 be the five forces. Draw two trans- versals Z, M intersecting A l9 ... A^. Construct the cylindroid of which Z, Mare the screws of zero pitch; find, upon this cylindroid, the screw X reciprocal to A 5 . Then the only movement which the body can receive, so as to fulfil the prescribed conditions, is a twist about the screw X. For X is then reciprocal to A l9 . . . A 5 , and therefore a body only free to twist about X will be Unacted upon by any forces directed along A^ . . . A 5 . From the theory of reciprocal screws it follows that a body rotated around any of the lines A ly . . . A s will not do work against nor receive energy from a wrench on X As a particular case, if A ly . . . A 5 have a common transversal, then X is that transversal, and its pitch is zero. In this case it is sufficiently obvious that A ly . . . A 5 cannot disturb the equilibrium of a body only free to rotate about X. 142. Impulsive Screws and Instantaneous Screws. A body which is free to twist about all the screws of a screw complex of the fourth order receives an impulsive wrench on the screw q. It is required to calculate the co-ordinates of the screw about which the body will commence to twist, and also the initial reactions of the constraints. Let A and /m be any two screws on the reciprocal DYNAMICS OF A RIGID BODY. 153 cylindroid, then the reaction of the constraints may be considered to consist of wrenches on X, /u of intensities X", fi". If we adopt the six absolute principal screws of inertia as the screws of reference, then the body will commence to move as if it were free, but had been acted upon by a wrench of which the co-ordinates are propor- tional to /A, . . ., / 6 6 . It follows that the given impul- sive wrench, when compounded with the reactions of the constraints, must constitute the wrench of which the co- ordinates have been just written ; whence if h be a cer- tain constant, we have the six equations 7, A n /' \ ff \ " npiVi = TI i]i + A Ai + ILL /ii, &C., &c. 7 A f) " i "\ "\ " np$* = TJ rj 6 + X A 6 -I- fj. ju 6 . Multiply the first of these equations by Xi, the second by X 2 , &c. : adding the six equations thus obtained, and observing that is reciprocal to X, we have = o, and similarly = O. From these two equations the unknown quantities X", ju" can be found, and thus the initial reaction of the constraints is known, substituting the values of X", ft" in the six original equations, the co-ordinates of the required screw are known. 143. Principal Screws of Inertia. We shall now show how the co-ordinates of the four principal screws of inertia belonging to the screw complex of the fourth order are to be computed. All the co-ordinates are, as before, referred to the six absolute principal screws of inercia of the body ( 105). Let c, j3 7, 8 be any four co-reciprocal screws of the 154 DYNAMICS OF A RIGID BODY. given screw complex. Then the co-ordinates of any other screw 6 of the complex may be determined by = a, -f + y + i, &c.,. &c. 0"0 6 = a"a G + |3"j3e + 7'V* + S"S 6 . We shall, as before, denote two screws on the reci- procal cylindroid by X, /u. If be a principal screw of inertia, then hp, ("ai + /3"j3i + 7'V + "'&) = a a, + /Tft + 77, + X'% + //'jui, &c., &c. A/. (a"a 6 + ]3"j3 6 + y">y. + S"&) = a'ae + jS'jSs + 7 " 7 6 + X"A 6 + /> 6 . Multiplying the first of these equations by ai, the next by a 2 , &c., adding the products, observing that a is reci- procal to j3, 7, 3, X, ju, and repeating the operations for |3, 7, 8, we have the four equations =o, A = o, -t- 7' / S iyi + From these four linear equations a", )3 /r , 7", 8" can be eliminated, and we obtain an equation of the fourth de- gree for h. When h is known, then a", j3", 7", 3" are known, and thus the co-ordinates of the four principal screws of inertia are determined. 144. Application of Euler's Theorem. It may be of interest to show how the instantaneous screw corres- ponding to a given impulsive sza2> ft, 72 />33 +JV/3 - z sft> Psft + 2 3 a 3 - ^373, ps7 3 + # 3 ft -J'sOs, ft, 73 ft, 7* ft, 75 By cyclical interchange the two analogous functions M and N are defined. The equation for p reduces to = o. The reduction of this equation to the first degree is an independent proof of the principle, that one screw, and only one, can be determined which is reciprocal to five given screws; p being known, a, /3, 7 can be found, and also two linear equations between .A/, y, 2', whence the reciprocal screw is completely determined. DYNAMICS OF A RIGID BODY. 163 152. Definition of the Sexiant. When six screws, A ly &c., At, are reciprocal to a single screw T, a certain relation must subsist between the six screws. This relation may be expressed by equating the determinant of 41 to zero. The determinant (called the sextant may be otherwise expressed as follows : The equations of the screw A k are a* /3* 7* We shall presently show that we are justified in assuming for T the equations = ~ = (pitch = p). o p 7 The condition that A k and T be reciprocal is (p 4- p*) (aa* + /3/3* + 77*) 4 Xk(yfik - fiyk) 4 yk(ayk - yak) + 2*(|3a* - a/3*) = O. Writing the six equations of this type, found by Iving k the values i to 6, and eliminating the six quantities pa, p/3, p7, a, /3, 7, we obtain the result : t 73^3 - + 75^5 ~ + ftp4 + ftps + ftpe + - 73*3, 73P3 + ~ 74^4, 74P4 + ~ 7s^5> 75P5 + ,, , 72 , ft, 73 4> ft, 7* s, ft, 7s e, ft, 7 6 By transformation to any parallel axes the value of this determinant is unaltered. The evanescence of the de- terminant is therefore a necessary condition whenever the six screws are reciprocal to a single screw. Hence we M2 1 64 DYNAMICS OF A RIGID BODY. sacrificed no generality in the assumption that 7"passed through the origin. Since the sexiant is linear in x^y^ z ly it appears that all parallel screws of given pitch reciprocal to one screw lie in a plane. Since the sexiant is linear in a : , /3i, ji we have Mobius' theorem ( 80). The property possessed by six screws when their sexiant vanishes may be enunciated in different ways,, which are precisely equivalent. (a). The six screws are all reciprocal to one screw. (b}. The six screws are members of a screw complex of the fifth order and first degree. (c}. Wrenches of appropriate intensities on the six screws equilibrate, when applied to a free rigid body. (d). Properly selected twist velocities about the six screws neutralize, when applied to a rigid body. (e). A body might receive six small twists about the- six screws, so that after the last twist the body would occupy the same position which it had before the first. If seven wrenches equilibrate (or twists neutralize), then the intensity of each wrench (or the amplitude of each twist) is proportional to the sexiant of the six non- corresponding screws. 153. Equilibrium. For a rigid body which has free- dom of the fifth order to be in equilibrium, the necessary and sufficient condition is that the forces which act upon the body constitute a wrench upon that one screw ta which the freedom is reciprocal. We thus see that it is not possible for a body which has freedom of the fifth order to be in equilibrium under the action of gravity unless the screw reciprocal to the freedom have zero pitch, and coincide in position with the vertical through the centre of inertia. Professor Sylvester has shown* that when six lines, * Comptes Rendus, tome 52, p. 816. See also p. 741. DYNAMICS OF A RIGID BODY. 1 65 P, Q, R, S y T, [7, are so situated that forces acting along them equilibrate when applied to a, free rigid body, a certain determinant vanishes, and the six lines are in involution* Using the ideas and language of the Theory of Screws, this determinant is the sexiant of the six screws, the pitches of course being zero. If ,T MJ , y my z m , be a point on one of the lines, the direc- tion cosines of the same line being a m , j3 m , 7, the condi- tion is - y\<*\ o. 02, p 2 , 72, J)Yy2-2 2 D 2 , 2 2 a 3 -# 2 7 2 , * 2 p 2 tt 4 , fii, 74, J) ; 474 24)84, 2 4 a 4 - -#474? #4J34 5> HO, 7 5 y^li^ ~ ^of^S? 25(15 X*TJ5) ^5f^a A single screw JT must be capable of being found which is reciprocal to all the six screws P, Q, R, S, T y U. Suppose the rigid body were only free to twist about X, then the six forces would not only collectively be in equilibrium, but severally would be unable to stir the body only free to twist about X. In general a body able to twist about six screws (of any pitch) would have perfect freedom ; but the body capable of rotating about each of the six lines, Py Qy Ry Sy Ty Uy which B.rQ in involution, is not ne- cessarily perfectly free (Mobius). * In the language of Pliicker (Neue Geometric des Raumes) a system of lines in involution forms a linear complex. In our language a system of lines in involution consists of the screws of equal pitch belonging to a screw complex of the fifth orden and first degree. See also Salmon's Geometry of Three Dimensions, third edition, p. 456, note. It may save the reader some trouble to observe here that the word involution has been employed in a more gene- ralised sense by Battaglini, and in quite a different sense by Klein. 1 66 DYNAMICS OF A RIGID BODY. If a rigid body were perfectly free, then a wrench about any screw could move the body ; if the body be only free to rotate about the six lines in involution, then a wrench about every screw (except X) can move it. The conjugate axes of Professor Sylvester (p. 743) are presented in the present system as follows : Draw any cylindroid which contains the reciprocal screw X, then the two screws of zero pitch on this cylindroid are a pair of conjugate axes. For a force on any transversal inter- secting this pair of screws is reciprocal to the cylindroid, and is therefore in involution with the original system. Draw any two cylindroids, each containing the re- ciprocal screw, then all the screws of the cylindroids form a screw complex of the third order. Therefore the two pairs of conjugate axes, being four screws of zero pitch, must lie upon the same quadric. This theorem is due to Professor Sylvester. The cylindroid also presents in a clear manner the solution of the problem of finding two rotations which shall bring a body from one position to any other given position. Find the twist which would effect the desired change. Draw any cylindroid through the corresponding screw, then the two screws of zero pitch on the cylindroid are a pair of axes that fulfil the required conditions. If one of these axes were given the cylindroid would be defined and the other axis would be determinate. 154. Impulsive Screws and Instantaneous Screws. We can determine the instantaneous screw corresponding to a given impulsive screw in the case of freedom of the fifth order by geometrical considerations. Let A, as before, represent the screw reciprocal to the freedom, and let p be the instantaneous screw which would correspond to A as an impulsive screw, if the body were perfectly free ; let r\ be the screw on which the body receives an DYNAMICS OF A RIGID BODY. 167 impulsive wrench, and let be the screw about which the body would commence to twist in consequence of this impulse if it had been perfectly free. The body when limited to the screw complex of the fifth order will commence to move as if it had been free, but had been acted upon by a certain unknown wrench on X, together with the given wrench on t|. The move- ment which the body actually acquires is a twisting motion about a screw which must lie on the cylindroid (, p). We therefore determine to be that one screw on the known cylindroid (, p) which is reciprocal to the given screw X. The twist velocity of the initial twisting motion about 9, as well as the intensity of the impulsive wrench on the screw X produced by the reaction of the constraints, are also determined by the same construc- tion. For by 17 the relative twist velocities about 0, , and p are known ; but since r/' is known, the twist velocity about is known ( 60) ; and therefore, the twist velo- city about is known ; finally, from the twist velocity about p, the intensity X" is determined. 155. Analytical Investigation. A quiescent rigid body which has freedom of the fifth order receives an impul- sive wrench on a screw r\ : it is required to determine the instantaneous screw 0, about which the body will com- mence to twist. Let X be the screw reciprocal to the freedom, and let the co-ordinates be referred to the absolute principal screws of inertia. The given wrench compounded with a certain wrench on X must constitute the wrench which, if the body were free, would make it twist about 0, whence we deduce the six equations (h being an unknown quantity). hpiQi = rf'tji -f X"Xi &c., &c., h$s = /V, + X"X C . 1 68 DYNAMICS OF A RIGID BODY. Multiplying the first of these equations by Xi, the second by X 2 , &c., adding the six equations thus produced, and remembering that 6 and X are reciprocal, we deduce 2 = o. This equation determines X" the intensity of the im- pulsive reaction of the constraints. The co-ordinates of the required screw 9 are, therefore, proportional to the six quantities rjiSXi 2 - Xi2?hXi. c ' - - ' &c - A 156. Principal Screws of Inertia. We are now ena- bled to determine the co-ordinates of the five principal screws of inertia ; for if be a principal screw of inertia, then whence with similar values for ? 2 , &c., 6 . Substituting these values in the equation and making = x y we have- for x the equation p\- x pi- x pi- x pi - x ps - x ps-x This equation is of the fifth d egree, corresponding to the five principal screws of inertia. If x' denote one of the roots of the equation, then the corresponding prin- cipal screw of inertia has co-ordinates proportional to AI Aj A 3 A 4 AS A DYNAMICS OF A RIGID BODY. 1 69 We may easily verify with these co-ordinates that each pair of principal screws of inertia are recipro- cal : for let xf, yd' be a pair of roots, then the differ- ence between the two equations = o and 2 - o is but this is equally the condition that the two screws of which the co-ordinates are shall be reciprocal ( 57). It is also easy to see that * ~ ' Since each of the terms on the left-hand side of this equation is zero, it follows that the right-hand side must be zero ; but this is equivalent ( 54) to the statement that the principal screws of inertia are conjugate screws of inertia ( 57). 170 CHAPTER XIV. THE DYNAMICS OF A RIGID BODY WHICH HAS FREEDOM OF THE SIXTH ORDER. 157. Introduction. When a rigid body has freedom of the sixth order, it is perfectly free. The screw com- plex of the sixth order includes every screw in space. That there is no reciprocal screw to such a complex is merely a different way of asserting the obvious pro- position that when a body is perfectly free it cannot remain in equilibrium, if the forces which act upon it have a resultant. 158. Impulsive Screws. Let A l9 A 2 , &c., denote a series of instantaneous screws which correspond re- spectively to the impulsive screws R l9 R 2 , &c., the body being perfectly free. Corresponding to each pair Ai 9 RI is a certain specific parameter. This parameter may be conveniently defined to be the twist velocity produced about At by an impulsive wrench on R i9 of which the intensity is one unit. If six pairs, AiRi, A Z R Z , &c., be known, and also the corresponding speci- fic parameters, then the impulsive wrench on any other screw R can be resolved into six impulsive wrenches on R 19 &c., R 6 , these will produce six known twist velocities on Ai 9 &c., A 6 , which being compounded together determine A, the twist velocity about A, and therefore the specific parameter of R and A. We thus see that it is only necessary to be given six corresponding pairs, and their specific parameters, in order to de- termine completely the effect of any other impulsive wrench. We are now going to show that z/ seven pairs a DYNAMICS OF A RIGID BODY. 1 7 1 corresponding instantaneous and impulsive screws be given, then the relation between every other pair is absolutely determined. It appears from 30 that appropriate twist velocities about A^ &c., A 1 can neutralise. When this is the case, the corresponding impulsive wrenches on JR l9 &c., R ly must equilibrate, and therefore the relative values of the intensities are known. It follows that the specific parameter of each pair A RI is proportional to the quotient obtained by dividing the sexiant of A 2 , &c., A^ by the sexiant of R z , &c., R 6 . With, therefore, the exception of a constant factor, the spe- cific parameter of every pair of screws is known, when seven corresponding screws are known. When therefore seven instantaneous screws are known, and the corresponding seven impulsive screws, we are enabled by geometrical construction alone to deduce the instantaneous screw corresponding to any eighth impulsive screw, and vice versa. A precisely similar similar method of proof will give us the following theorem : If a rigid body be in position of stable equilibrium under the influence of a sytem of forces which have a potential, and if the twists abont seven given screws evoke wrenches about seven other given screws, then, without knowing any further about the forces, we shall be able to determine the screw on which a wrench is evoked by a twist about any eighth screw. We shall state the results of the present section in a form, which may, perhaps, interest the student of mo- dern geometry. We must conceive two corresponding systems of screws, of which the correspondence is com- pletely established, when, to any seven screws regarded as belonging to one system, the seven corresponding screws in the other system are known. To every screw in space viewed as belonging to one system will corres- 172 DYNAMICS OF A RIGID BODY. pond another screw viewed as belonging to the other system. Six screws can be found, each of which coin- cides with its correspondent. To a screw complex of the n th order and m th degree in one system will corres- pond a screw complex of the n th order and m th degree in the other system. We add here a few examples to illustrate the use which may be made of screw co-ordinates. 159. Theorem. When an impulsive force acts upon a free quiescent rigid body, the directions of the force and of the instantaneous screw are parallel to a pair of conjugate diameters in the momental ellipsoid. Let 7i, ... i7s be the co-ordinates of the force referred to the absolute principal screws of inertia, then (37) (in + n) 2 + (-773 + m) 8 + (r? 5 + r/ 6 ) 2 = i, and from ( 93) it follows that the direction cosines of 17 with respect to the principal axes through the centre of inertia are (h + *?s)> (la + m), (i?5 + fc). If a, b f c be the radii of gyration, then the instan- taneous screw corresponding to 17 has for co-ordinates The condition that 17 and its instantaneous screw shall be parallel to a pair of conjugate diameters of the mo- mental ellipsoid is or But if the impulsive wrench on 17 be a force, then the pitch of 17 is zero, whence the theorem is proved. DYN AMICS OF A RIGID BODY. 173 1 60. Theorem. When an impulsive wrench acting on a free rigid body produces an instantaneous rotation, the axis of the rotation must be perpendicular to the im- pulsive screw. Let ni, . . . rj 6 be the axis of the rotation, then or whence the screw of which the co-ordinates are + - 0rj>, + bv} Zy &c., is perpendicular to r/, and the theorem is proved. From this theorem, and the last, we infer that, when an impulsive force acting on a rigid body produces an instantaneous rotation, the direction of the force, and the axis of the rotation, are parallel to the principal axes of a section of the momental ellipsoid. 161. Principal Axis. If rj be a principal axis of a rigid body, it is required to prove that reference being made to the absolute principal screws of inertia. For in this case a force along a line 6 intersecting /, compounded with a couple in a plane perpendicular to r\, must constitute an impulsive wrench to which rj corres- ponds as an instantaneous screw, whence we deduce ( 93)> ^ an d k being arbitrary constants. . l -7- -T !!, P\ am &c., n h dR ^ 06 = -r -T + kpwt. pt dm Expressing the condition that pe = o, we have 1 74 DYNAMICS OF A RIGID BODY. -T- + A 2 2 i -T- = o ; but we have already seen ( 93, 105) that the two last terms of this equation are zero, whence the required theorem is demonstrated. The formula we have just proved may be written in the form 2/i . /irji . pirn = O. This shows that if the body were free, then an impulsive force suitably placed would make the body commence to rotate about ?j. Whence we have the following" theorem : A rigid body previously in unconstrained equilibrium in free space is supposed to be set in motion by a single impulsive force ; if the initial axis of twist velocity be a principal axis of the body, the initial motion is a pure rotation, and conversely. (Mr. Townsend, Educational Times, reprint, Vol. xxi., p. 107.) It may also be asked what is the point of the body one of the three principal axes through which coincides with rj ? This point is the intersection of and rj. To determine the co-ordinates of 9 it is only necessary to find the relation between h and k, and this is obtained by expressing the condition that is reciprocal to rj, whence we deduce 2h 4 kuj = o. Thus 6 is known, and the required point is determined. If the body be fixed at this point, and then receive the impulsive couple perpendicular to 17, the instantaneous reaction of the point will be directed along 9. 1 62. Harmonic Screws. We shall conclude by stating vfor the sixth order the results which are included as par- ticular cases of the general theorems in Chapter VIII. DYNAMICS OF A RIGID BODY. 175 If a perfectly free rigid body be in equilibrium under the influence of a system of forces as restricted in 6, then six screws can be found such that each pair are conjugate screws of inertia, as well as conjugate screws of the potential, and these six screws are called harmonic screws. If the body be displaced from its position of equilibrium by a twist of small amplitude about a harmonic screw, and if the body further receive a small initial twisting motion about the same screw, then the body will continue for ever to perform small twist oscillations about that screw. And, more generally, whatever be the initial circum- stances, the movement of the body is compounded of twist oscillations about the six harmonic screws. APPENDIX, No. I. I HERE briefly describe the principal works known to me which bear on the subject of the present volume. POINSOT (L.) Sur la composition des moments et la composition des aires (1804). Paris Journal de 1'Ecole Polytechnique ; t. vi. 13 cah., pp. 182-205 (1806). In this paper the author of the conception of the couple, and of the laws of composition of couples, has demonstrated the important theorem that any system of forces applied to a rigid body can be reduced to a single force, and a couple in a plane perpen- dicular to the force. CHASLES (M.) Note sur les proprietes generales du systeme de deux corps semblables entfeux et places d'une maniere quelconque dans Vespace; et sur le deplacement fini ou infiniment petit d?un corps solide libre. Bulletin des Sciences Mathe*ma- tiques, par Ferussac. Vol. xiv., pp. 321-326 (Paris, 1830). The author shows that there always exists one straight line, about which it is only necessary to rotate one of the bodies to place it similarly to the other. Whence (p. 324) he is led to the following fundamental theorem : Eon peut ioujours transporter un corps solide libre d'une position dans une autre position quelconque, determime par le mouvement con- tinu d'une vis a laquelle ce corps serait fixk invariablement . N 178 APPENDIX I. That this theorem is really due to Chasles there can be little doubt. He explicitly claims it in note 34 to the Aperqu Histo- rique. Three or four years later'than the paper we have cited, Poinsot published his celebrated " Theorie Nouvelle de la Rota- tion des Corps" (Paris, 1834). In this he enunciates the same theorem. As Poinsot does not refer to Chasles, I had been led, in ignorance of Chasles' previous paper, to attribute the theorem to Poinsot (Transactions of Royal Irish Academy, Vol. xxv., p. 1 60); but I corrected the mistake in Phil. Trans., 1874, p. 16. MOBIUS (A. F.) Lehrbuch der Statik (Leipzig, 1837). This book is, we learn from the preface, one of the numerous productions to which the labours of Poinsot has given rise. The first part, pp. 1-355, discusses the laws of equilibrium of forces, which act upon a single rigid body. The second part, pp. 1-313, discusses the equilibrium of forces acting upon several rigid bodies connected together. The charac- teristic feature of the book is its great generality. I here enunciate some of the principal theorems. If a number of forces acting upon a free rigid body be in equilibrium, and if a straight line of arbitrary length and position be assumed, then the algebraic sum of the tetrahedra, of which the straight line and each of the forces in succession are pairs of opposite edges, is equal to zero (p. 94). If four forces are in equilibrium they must be generators of the same hyperboloid (p. 177). If five forces be in equilibrium they must intersect two common straight lines (p. 179). If the lines of action of five forces be given, then a certain plane S through any point P is determined. If the five forces can be equilibrated by one force through P, then this one force must lie in S (p. 180). To adopt the notation of Professor Cayley, we denote by 12 the perpendicular distance between two lines i, 2, multiplied into the sine of the angle between them (Comptes Rendus, t. Ixi., pp. 829-830). Mobius shows (p. 189) that if forces APPENDIX I. 179 along four lines i, 2, 3, 4 equilibrate, the intensities of these forces are proportional to ^23.24.14, ^13.14.34, A/12. 14. 24, A/12. 13.23 It is also shown that the product of the forces on i and 2, multiplied by 12, equals the product of the forces on 3 and 4 multiplied by 34. He hence deduces Chasles' theorem (Liou- ville's Journal, t. xii., p. 222), that the volume of the tetrahe- dron formed by two of the forces is equal to that formed by the remaining two. i MoBlUS (A. F.) Ueber die Zusammensetzung unendlich kleiner Drehungen. Crelle's Journal ; t. 18, pp. 189-212 (Berlin, 1838). This memoir contains many very interesting theorems, of which the following are the principal : Any given displace- ment of a rigid body can be effected by two rotations. Two equal parallel and opposite rotations compound into a translation. Rotations about intersecting axes are compounded like forces, If a number of forces acting upon a free body make equilibrium, then the final effect of a number of rotations (proportional to the forces) on the same axes will be zero. If a body can be rotated about six independent axes, it can have any movement whatever. RODRIGUES (O.) Des lots geometriques quirtgissent les deplacements d'unsysteme solide dans Fespace et de la variation des co-ordon- nees, provenant de ces deplacements considers independamment des causes qui peuvent les produire. Liouville's'Journal ; t. 5, pp. 380-440 (5th Dec., 1840). This paper consists mainly of elaborate formulae relating to displacements of finite magnitude. It has been already cited for an important remark ( 12). 180 APPENDIX I. CHASLES (M.) Proprietes geometriques relatives au mouvement in- finiment petit cTun corps solide libre dans. Vespace. Comptes Rendus; t. xvi., pp. 1420-1432 (1843). A pair of " droites conjugue"es" are two lines by rotations about which a given displacement can be communicated to a rigid body. Two pairs of " droites conjugue"es" are always generators of the same hyperboloid. POINSOT (L.) Th&orie nouvelle de la rotation des corps. Liouville's Journal ; t. xvi., pp. 9-129, 289-336 (March, 1851). This is Poinsot's classical memoir, which contains his beau- tiful geometrical theory of the rotation of a rigid body about a fixed point. In a less developed form the Theory had been pre- viously published in Paris in 1834, as already mentioned. CAYLEY (A.) On a new analytical representation of curves in space Quarterly Mathematical Journal; Vol. iii., pp. 225-236 (1860). Vol. v., pp. 81-86. In this paper the conception of the six co-ordinates of a line is introduced for the first time. SYLVESTER (J. J.) Sur T involution des lignei droites dans Tespace* considerees comme des axes de rotation. Comptes Rendus ; t. Hi., pp. 741-746 (April, 1861). Any small displacement of a rigid body can generally be represented by rotations about six axes (Mobius). But this is not the case if forces can be found which equilibrate when acting along the six axes on a rigid body. The six axes in this case are in involution. The paper discusses the geometrical features of such a system, and shows, when five axes are given, how the locus of the sixth is to be found. Mobius had shown that through any point a plane of lines can be drawn in involu- tion with five given lines. The present paper shows how the plane can be constructed. All the transversals intersecting a pair of conjugate axes are in involution with five given lines. Any two pairs of conjugate axes lie on the same hyperboloid. APPENDIX I. l8l Two forces can be found on any pair of conjugate axes, which are statically equivalent to two given forces on any other given pair of conjugate axes. In presenting this paper M. Chasles remarks that Mr. Sylvester's results lead to the following construction : Conceive that a rigid body receives any small displacement, then lines drawn tnrough any six points of the body perpendicular to their trajectories are in involution. M. Chasles takes occasion to mention also some other properties of the conjugate axes. SYLVESTER (J. J.) Note sur V involution de six lignes dans Vcspacf. Comptes Rendus; t. Hi., pp. 815-817 (April, 1861). The six lines are i, 2, 3, 4,5, 6. Let the line i be repre- sented by the equations + c& + d-u = o, o-iX + fry + y 3 i, 4 i, 5 i, 6 2, I 2, 3 2, 4 2, 5 2, 6 3, i 3, 2 3, 4 3 5 3, 6 4, i 4 2 4> 3 4 5 4 6 5, i 5, 2 5 3 5, 4 5. 6, i 6, 2 6, 3 6, 4 6, 5 If A 6 = o, the lines are in involution. Considering only the figures i, 2, 3, 4, 5, the determinant A a can be formed. If 1 82 APPENDIX I. A, = o and A, = o, the five lines i, 2, 3, 4, 5 are in involution. If all the other minors are zero, the six lines will intersect a single transversal. If A 5 = o, without any other condition, the five lines i, 2, 3, 4, 5 intersect a single transversal. If A 4 = o without any other condition, the lines i, 2, 3, 4 have but one common transversal (Cayley). A determinant can be found which is equal to the square root of A.. This square root is the determinant given in 153. GRASSMANN (H.) Die Ausdehnungskhre. Berlin (1862). A system of n, numerically equal, " Grossen erster Stufe," of which each pair are " normal," is discussed on p. 113. A set of co-reciprocal screws is a particular case of this very general conception. The "inneren Produkte" of two "Grossen" divided by the product of their numerical values, is the cosine of the angle between the two " Grossen." If a, I, c, . . . be normal, and if k, I be any two other " Grossen," then cos Lkl = cos Lak costal + cos Lbk. cosZ./, +&c. (p. 139). Here we have a very general theory, which includes screw co-ordinates as a particular case. In a note on p. 222 the author states that the displacement of a body in space, or a general system of forces, form an " allge- meine raumliche Grosse zweiter Stufe." The " kombinatorische Produkt" (p. 41) of n screws will contain as a factor that single function whose evanescence would express that the n screws belonged to a screw com- plex of the (n-ij h order. PLUCKER (J.) On a new geometry of space. Phil. Trans., 1865. Vol. 155, pp. 725-791. In this paper the linear complex is defined (p. 733). Some applications to optics are made (p. 760) ; the six co-ordinates of a line are considered (p. 774) ; and the applications to the geometry offerees (p. 786). APPENDIX I. PLUCKKR (J.) Fundamental views regarding mechanics. Phil. Trans. (1866), Vol. 156, pp. 361-380. The object of this paper is to " connect, in mechanics, translator/ and rotatory movements with each other by a princi- ple in geometry analogous to that of reciprocity." One of the principal theorems is thus enunciated: " Any number of rota- tory forces acting simultaneously, the co-ordinates of the result- ing rotatory force, if there is such a force, if there is not, the co-ordinates of the resulting rotatory dyname, are obtained by adding the co-ordinates of the given rotatory forces. In the case of equilibrium the six sums obtained are equal to zero." MANNHEIM (A.) Sur le dlplacement cTun corps solide. Journal de Mathe"matiques, 2" Series, t. xl. (1866). To M. Mannheim belongs the credit of having been the first to study geometrically the kinematics of a constrained rigid body from a perfectly general point of view. This paper con- tains the following theorem : When a rigid body has freedom of the second order, any point of the body must be displaced on a certain surface, and at any instant all the normals to these surfaces will intersect two straight lines. This is easily seen from the Theory of Screws, because any force reciprocal to the cylindroid expressing this freedom must be a normal to all the surfaces belonging to the points on it. SPOTTISWOODE (W.) Note sur Vequilibre des forces dans fespace. Comptes Rendus; t. Ixvi., pp. 97-103 (January, 1868). If P , &c., P n _i be n forces in equilibrium, and if (o, i) denote the moment of P , PI, then the author proves* that P^o, i) + P f (o, 2) + &c. = o, P(i, o) + + P t (i, 2) + &c. = o, P(2 t o) + P l (2, l)+ +. ..-O. * We may remark that since the moment of two lines is the virtual co-effi- cient of two screws of zero pitch, these equations are given at once by virtual velocities, if we rotate the body round each of the forces in succession. 1 84 APPENDIX I. As we have thus n equations to determine only the relative values of n quantities, the redundancy is taken advantage of to prove that z? z?t = &C., [0,0] [i, i] where [o, o], [i, i], &c., are the coefficients of (o, o), (i, i), &c., in the determinant (o, o) (o, i) . . . (i, o) (i, i) ... When the number of forces is less than seven, it is shown how the formula? admit of a special transformation, which expresses the conditions to be fulfilled. This very elegant result may receive an extended interpreta- tion. If P , /> P 2 , &c., denote the intensities of wrenches on the screws o, i, 2, &c. ; and if (12) denote the virtual co-efficient of i and 2, then, when the formulae of Mr. Spottiswoode are satis- fied, the n wrenches equilibrate, provided that the screws belong to a screw complex of the (n - ij h order and first degree. PLUCKER (J.) Neue Geometric des Raumes gegriindet auf die Betrachtung der geraden Linie ah Raumelement. Leipzig (B. G. Triibner, 1868-69), PP- J ~374- This elaborate work is the principal authority on the theory of the linear complex. The subject is essentially geometrical, but there are a few remarks on mechanics ; thus the author, on p. 24, introduces the word " Dyname :" " Durch den Ausdruck 'Dyname/ habe ich die Ursache einer beliebigen Bewegung eines starren Systems, oder, da sich die Natur dieser Ursache, wie die Natur einer Kraft iiberhaupt, unserem Erkennungsver- mogen entzieht, die Bewegung selbst : statt der Ursache die Wirkung, bezeichnet." Although it is not very easy to see the precise meaning of this passage, yet it appears that a ' Dyname J APPENDIX I. 185 may be either a twist or a wrench (to use the language of the Theory of Screws.) On p. 25 we read : " Dann entschwindet das specifisch Mechanische, und, um mich auf eine kurze Andeutung zu be- schranken : es treten geometrische Gebilde auf, welche zu Dyna- men in derselben Beziehung stehen, wie gerade Linien zu Kraften und Rotationen." There can be little doubt that the " geometrische Gebilde," to which Pliicker refers, are what we have called screws. As we have already stated ( 16), it is in this book that we find the first mention of the surface which we call the cylindroid. Through any point a cone of the second degree can be drawn, the generators of which are lines belonging to a linear complex of the second degree. If the point be limited to a certain sur- face the cone breaks up into two planes. This surface is of the fourth class and fourth degree, and is known as Rummer's sur- face, or as the surface of singularities appropriate to the given linear complex. (See Kummer, Abhandl. der Berl. Akad., 1866). This theory is of interest for our purpose, because the locus of screws reciprocal to a cylindroid is a very special linear complex of the second degree, of which the cylindroid itself is the surface of singularities. KLEIN (Felix). Zur Theorie der Linien- Complex e des ersten und zweiten Grades. Math. Ann., II. Band, pp. 198-226 ( i4th June, 1869). The "simultaneous invariant" of two linear complexes is discussed. In our language this function is the virtual co- efficient of the two screws reciprocal to the complexes. The six fundamental complexes are considered at length, and many remarkable geometrical properties proved. It is a matter of no little interest that these purely geometrical re- searches have a physical significance attached to them by the Theory of Screws. This paper also contains the following proposition: If ,* . . ., x t be the co-ordinates of a line, and k vt ... k t be con- 1 86 APPENDIX I. stants, then the family of linear complexes denoted by x? x* r + &C. + j-i-r = O, RI - A K 6 - A have a common surface of singularities where X is a variable parameter. If the roots A,, &c. be known, we have a set of quasi elliptical co-ordinates for the line x. (Compare with 156). It is in this memoir that we find the enunciation of the remarkable geometrical principle which, when transformed into the language and conceptions of the Theory of Screws, asserts the existence of one screw reciprocal to five given screws. KLEIN (F.) Die allgemeine lineare Transformation der Linien- Co-ordinaten. Math. Ann., Vol. ii., p. 366-371 (August 4,. 1869). Let U v , . . U, denote six linear complexes. The moments of a straight line, with its conjugate polars with respect to U^ ... 7,, are, when multiplied by certain constants, the homogeneous co-ordinates of the straight line, and are denoted by x it . . . x t . Arbitrary values of x lt &c., do not denote a straight line, unless a homogeneous function of the second degree vanishes.* If this condition be not satisfied, then a linear complex is defined by the co-ordinates, and the function is called the invariant of the linear complex. The simultaneous invariant of two linear com- plexes is a function of the co-ordinates, or is equal to A sin - (K + K 1 ) cos <, where K and K are the parameters of the linear complexes, A the perpendicular distance, and the angle between their principal axes. If this quantity be zero, the two linear complexes are in involution. (The reader will observe that the word involution is here employed in a very different sense to that in which the same word is used by Professor Sylvester.) The co-ordinates of a linear complex are the simultaneous * This equation expresses that the pitch of the screw denoted by the co-ordinates is zero. APPENDIX I. 187 invariants of the linear complex with each of six given linear complexes multiplied by certain constants. The six linear com- plexes can be chosen so that each one is in involution with the remaining five. The reader will easily perceive the equivalent theorems in the Theory of Screws: ZEUTHEN (H. G.} Notes sur un systems de co-ordonnees liniaire dans fespace. Math. Ann., Vol. i., pp. 432-454 (1869). The co-ordinates of a line are the components of an unit force on the line decomposed along the six edges of a tetrahedron. These co-ordinates must satisfy one condition, which expresses that six forces along the edges of a tetrahedron have a single resultant force. The author makes applications to the theory of the linear complex. Regarding the six edges as screws of zero pitch, they are not co-reciprocal. It may, however, be of interest to show how these co-ordinates may be used for a different purpose from that for which the author now quoted has used them. Call the virtual co-efficients of the opposite pairs of edges Z, M, N. If the co-ordinates of a screw with respect to this system be 0! . . . a , then the pitch is and the virtual co-efficient of the two screws <, 6 is L BATTAGLINI (G.) Memoria sulk dinami in involuzione. Atti di Napoli IV. (1869). The co-ordinates of a dyname are the six forces which acting along the edges of a tetrahedron, are equivalent to the dyname. This memoir investigates the properties of dynames of which the co-ordinates satisfy one or more linear equations. The author shows analytically the existence of two associated systems of dynames such that all the dynames of the first order are correlated to all the dynames of the second. These correspond to what we would call two reciprocal screw com- plexes. 1 88 APPENDIX I. BATTAGLINI (G.) Sul movimento geometrico infinitesimo di un sistemo rigido. Estratto dal Rendiconto della JR. Accademia delle Seienze fisiche e Matematiche. (Fascicolo, May 5, 1870). This paper and the last belong to a series by the same author, in which the tetrahedron co-ordinates are employed in the analytical development of the statics of a rigid body, as well as the theory of small displacements. MANNHEIM (A.) Etude sur h deplacement dune figure de forme in- variable. Recueil des Memoires des Savants etrangers ; t. xx. Journal de l'6cole Polytechnique, cah. 43, pp. 57-122 (1870). This paper discusses the trajectories of the different points of a body when its movement takes place under prescribed condi- tions. Had I been sooner acquainted with this paper, I should have attributed to M. Mannheim the theorem about the screws of zero pitch on a cylindroid given in 95. Another theorem of the same class is also given by M. Mannheim. When a rigid body has freedom of the third order, then for any point on the surface of a certain quadric* the possible displacements are limited to a plane. BALL (R. S.) On the small oscillations of a Rigid Body about a fixed point under the action of any forces, and, more particu- larly, when gravity is the only force acting. Transactions of the Royal Irish Academy, Vol. xxiv., pp. 593-627 (January 24, 1870.) The principal theorems contained in this paper are demon- strated in 130 of the present volume. * The reader \rill easily see that this is the pitch quadric. APPENDIX I. 189 KLEIN (Felix). Notiz bttreffend den Zusammenhang der Linien- Geomdrie mit der Mechanik starrer Korper. Math. Ann., Vol. iv., pp. 403-415 (June, 1871). Among many interesting matters this paper contains the germ of t\ie physical conception of reciprocal screws. We thus read on p. 413: "Es lasst sich nun in der That' ein physikalischer Zusammenhang zwischen Kraftesystemen und unendlich kleinen Bewegungen angeben, welcher es erklart, wie so die beiden Dinge mathematisch co-ordinirt auftreten. Diese Beziehung ist nicht von der Art, dass sie jedem Kraftesystem eine einzelne unendlich kleineBewegungzuordnet, sondern sie ist von anderer Art, sie ist eine dualistische. " Es sei ein Kraftesystem mit de^i Coordinaten E, ff, Z, A, M, N, und eine unendlich kleine Bewegung mit den Coordi- naten ', H', Z' y A', M 7 , ^gegeben, wobei man die Co-ordinaten in der im 2 besprochenen Weise absolut bestimmt haben mag. Dann reprdsentirt, wie hier nicht weiter nachgewiesen werden soil, der Ausdruck A'E + M'N+ WZ + E'A + ITM+ Z'N das Quantum von Arbeit, welches das gegebene Kraftesystem bei Eintritt der gegebenen unendlich kleinen Bewegung leistet. Ist insbesondere A'E + MH + JSTZ + E'A + ITM+ Z'N= o, so leistet das gegebene Kraftesystem bei Eintritt der gegebenen unendlich kleinen Bewegung keine Arbeit. Diese Gleichung nun reprasentirt uns, indem wir einmal E, H, Z, A, M, N, das andere E', H', Z', A', M', N' als veranderlich betrachten, den Zusammenhang zwischen Kraftesystemen und unendlich kleinen Bewegungen." KLEIN (Felix). Ueber gewtsse in der Linien-Geometrie auftretende Differential- Gleichungen. Math. Ann., V. Band, pp. 278 -303 (November, 1871). There is a remarkable invariant of n linear complexes #i * o, U* = o, . . . U n = o. For let AI, . . . A,. be arbitrary multipliers, 190 APPENDIX I. then \\U\ + . . . + A n 7 M = o also denotes a linear complex, pro- vided that a certain condition is satisfied. This condition is presented as a homogeneous function of the second degree in A!, . . -. A w equated to zero. The discriminant of the function is the invariant in question. BALL (R. S.) The Theory of Screws a geometrical study ofihe kinematics, equilibrium, and small oscillations of a Rigid Body. Transactions of the Royal Irish Academy, Vol. xxv., pp. 137-217 (November 13, 1871). This is the original paper on the Theory of Screws. In estimating how far the contents of this paper are novel, it is to be remembered that the cylindroid had been discussed by Pliicker two or three years previously, while the conception of reciprocal screws had been announced by Klein a few months before. Both these authors would, of course, have been re- ferred to in this paper had I been acquainted with their works at the time the paper was written. CLIFFORD (W. K.) On Biquaternions. Proceedings of the Lon- don Mathematical Society, Nos. 64, 65, p. 382 (i2th June, 1873). A Biquaternion is defined to be the ratio of two " motors." A "motor" may be said to bear the same relation to the dyname of Pliicker which a vector bears to a linear magnitude. The Biquaternions are shown to be intimately associated with the speculations of the geometry of elliptic space. See Klein's wonderful paper, " Ueber die nicht Euclidische Geometric." Math. Ann., Band IV., pp. 573-625. BALL (R. S.) Researches in the Dynamics of a Rigid Body by the aid of the Theory of Screws (June 19, 1873). Philosophical Transactions, pp. 15-40 (1874). The n principal screws of inertia belonging to a rigid body which has freedom of the n ih order are here discussed. APPENDIX I. 191 LINDEMANN (F.) Ueber unendlich kleine Bewegungen und uber Krdftesysteme bet allgemeiner projectivischer Massbestimmung. Math. Ann., yth Vol., pp. 56-143 (July, 1873). This is a memoir upon the statics and kinematics of a rigid body in elliptic or hyperbolic space. Among several results closely related to the Theory of Screws, we find that the cylin- droid is only the degraded form in parabolic or common space of a surface of the fourth order, with two double lines. WEILER (A.) Ueber die verschiedenen Gattungen der Complex* zweiten Grades. Math. Ann., Vol. vii., pp. 145-207 (July, 1873). In this elaborate memoir the author enumerates fifty-eight different species of linear complex of the second order. The classification is based upon Rummer's surface, which defines the singularities of the screw complex. BALL, (R. S.) Screw Co-ordinates and their applications to pro- blems in the Dynamics of a Rigid Body. Transactions of the Royal Irish Academy, Vol. xxv., pp. 259-327 (January 12, 1874). To trace a satisfactory connexion between an impulsive screw and the corresponding instantaneous screw is the principal object of this paper. It is here shown that to the instantaneous screw, whose co-ordinates are Oi t . . . , 6 , corresponds an impul- sive screw, whose co-ordinates are proportional to /A, . . ., / 8 0, reference being made to the absolute principal screws of inertia. EVERETT (J. D.) On a new method in Statics and Kinematic*. (Part I.) Messenger of Mathematics. New Series. No. 39 (1874). This paper contains applications of quaternions. The opera- tor v + Var ( ) is a " motor," and o- being vectors, the former denoting a translation or couple, the latter a rotation or force. The pitch is S-. The equation to the central axis is p = P- cr a - p$ (in the present case o?o66) is divided into one hundred parts. By the screw, which moves the slide rest parallel to the bed of the lathe, the drill can be moved to any number z of these parts from its original position at the centre of the length of the cylinder. Four holes are to be drilled for each value of z. These consist of two pairs of diametrically opposite holes. The directions of the holes intersect the axis of the cylinder at right angles. The following table will enable the work to be executed with facility. / is the angle of 15 : z / go-/ r8o + / 270 / O.O 90 i So 70 17-4 5 85 .85 265 34'2 10 80 190 260 5O'O 15 75 *95 2 55 64-3 20 70 200 250 76-5 25 65 205 245 86-6 30 60 2IO 240 94-0 35 55 215 235 98-5 40 5 22O 230 100-0 45 45 225 225 IQ4 APPENDIX II. For example, when the slide has been moved 34-2 parts from the centre of the cylinder, the dividing plate is to be set successively to 10, 80, 190, 260, and a hole is to be drilled in at each of these positions. The slide rest is then to be moved on to 50 parts, and holes are to be drilled in at 15, 75, 195, 255. Steel wires, each about 0^3 long, are to be forced into the holes thus made, and half the surface is formed. The remaining half can be similarly constructed : a length of o?c>66 cos 2/ is to be coloured upon each wire to show the pitch. The sign of the pitch is indicated by using one colour for positive, and another colour for negative pitches. THE END. 5 8606