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Full text of "A treatise on the theory of screws"




HATH-STAJ. 



I 



A TEBATISE 



ON THE 



THEOBY OF SOEEWS, 



: C. J. CLAY AND SONS, 
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 

AVE MAKTA LANE. 
lasgofo: 50, WELLINGTON STREET. 




ILetpjig : F. A. BBOCKHAUS. 
gork: THE MACMILLAN COMPANY. 
Bombnp: E. SEYMOUR HALE. 




THE CENTRAL PORTION OF THE CYLINDROID. 



Frontispiece. 



A TREATISE 



ON THE 



THEORY OF SCREWS 



SIR ROBERT STAWELL BALL, LL.D., F.R.S., 

LOWNDEAN PEOFESSOE OF ASTEONOMY AND GEOMETEY IN THE 

TTNIVEESITY OF CAMBEIDGE, 
FOEMEELY EOYAL ASTEONOMEE OF IEELAND. 



CAMBRIDGE: 
AT THE UNIVERSITY PRESS. 

1900 

[All Rights resented.] 



Or 4^.. 



MATH-STAT. 



PRINTED BY J. AND C. F. CLAY, 

AT THE UNIVERSITY PRESS. 



MATtU 

STAT. 

UBRARY 



PEE FACE. 



A BOUT thirty years ago I commenced to develop the consequences of 
-*- certain important geometrical and dynamical discoveries properly 
associated with the illustrious names of Poinsot and Chasles, Hamilton 
and Klein. The result of rny labours I have ventured to designate as 
" The Theory of Screws." 

As the theory became unfolded I communicated the results in a long 
series of memoirs read chiefly before the Royal Irish Academy. To 
this learned body I tender my grateful thanks for the continual kind 
ness with which they have encouraged this work. 

I published in 1876 a small volume entitled The Theory of Screws: 
A Study in the Dynamics of a Rigid Body. This contained an account 
of the subject so far as it was then known. 

But in a few years great advances were made, the geometrical 
theories were much extended, and the Theory of Screw-chains opened 
up a wide field of exploration. The volume just referred to became 
quite out of date. 

A comprehensive account of the subject as it stood in 1886 was 
given in the German work Theoretische Mechanik starrer Systeme : Auf 
Grund der Methoden und Arbeiten und mit einem Vorworte von Sir Robert 
S. Ball, herausgegeben von Harry Gravelius, Berlin, 1889. This work was 
largely a translation of the volume of 1876 supplemented by the sub 
sequent memoirs, and Dr Gravelius made some further additions. 

The theory was still advancing, so that in a few years this considerable 
volume ceased to present an adequate view of the subject. For example, 
the Theory of Permanent Screws which forms perhaps one of the most 
B. b 



M777321 



VI PREFACE. 

instructive developments was not communicated to the Royal Irish Academy 
until 1890. The twelfth and latest memoir of the series containing the 
solution of an important problem which had been under consideration 
for twenty-five years did not appear until 1898. 

It therefore seemed that the time had now come when an attempt 
should be made to set forth the Theory of Screws as it stands at 
present. The present work is the result. I have endeavoured to include 
in these pages every essential part of the Theory as contained in the 
twelve memoirs and many other papers. But the whole subject has 
been revised and rearranged and indeed largely rewritten, many of the 
earlier parts have been recast with improvements derived from later 
researches, and I should also add that I have found it necessary to 
introduce much that has not been previously published. 

The pleasant duty remains of expressing my thanks for the help that 
I have received from friends in preparing this book. I have received 
most useful aid from Prof. W. Burnside, Mr A. Y. G. Campbell, 
Mr G. Chawner. Mr A. W. Panton, Mr H. W. Richmond, Mr R. Russell, 
and Dr G. Johnstone Stoney. In the labour of revising the press I have 
been aided by Mr A. Berry, Mr A. N. Whitehead, and lastly by Professor 
C. J. Joly, who it will be seen has contributed several valuable notes. 

Finally, I must express my hearty thanks to the Cambridge Univer 
sity Press for the liberality with which they undertook the publication 
of this book and for the willing consent with which they have met 
all my wishes. 

ROBERT S. BALL. 



OBSERVATORY, 
CAMBRIDGE, 17 May, 1900. 



CONTENTS. 



PAGE 

INTRODUCTION . 1 



CHAPTER I. 

TWISTS AND WRENCHES. 

1. Definition of the word Pitch ......... 6 

2. Definition of the word Screw 7 

3. Definition of the word Twist ......... 7 

4. A Geometrical Investigation .......... 8 

5. The canonical form of a Small Displacement ...... 9 

6. Instantaneous Screws 10 

7. Definition of the word Wrench 10 

8. Restrictions . . 11 



CHAPTER II. 
THE CYLTNDROID. 

9. Introduction 15 

10. The Virtual Coefficient 17 

11. Symmetry of the Virtual Coefficient ........ 18 

12. Composition of Twists and Wrenches 18 

13. The Cylindroid . 19 

14. General Property of the Cylindroid 21 

15. Particular Cases 22 

16. Cylindroid with One Screw of Infinite Pitch 22 

17. Form of the Cylindroid in general 24 

18. The Pitch Conic 24 

19. Summary 24 

62 



Vlll CONTENTS. 

CHAPTER III. 

RECIPROCAL SCREWS. 








PAGE 


20. 


Reciprocal Screws ............ 


26 


21. 


Particular Instances 


26 


22. 


Screw Reciprocal to Cylindroid ......... 


26 


23. 


Reciprocal Cone ............ 


27 


24. 


Locus of a Screw Reciprocal to Four Screws ...... 


29 


25. 


Screw Reciprocal to Five Screws ......... 


30 


26. 


Screw upon a Cylindroid Reciprocal to a Given Screw .... 


30 


27. 


Properties of the Cylindroid .......... 


30 




CHAPTER IV. 






SCREW CO-ORDINATES. 




28. 


Introduction 


31 


29. 


Intensities of the Components ......... 


32 


30. 


The Intensity of the Resultant 


33 


31. 


Co-reciprocal Screws 


33 


32. 


Co-ordinates of a Wrench .......... 


34 


33. 


The Work done 


34 


34. 


Screw Co-ordinates 


34 


35 




35 


36. 


Calculation of Co-ordinates .......... 


35 


37. 


The Virtual Coefficient 


36 


38 


The Pitch 


36 


39. 


Screw Reciprocal to Five Screws ......... 


36 


40. 


Co-ordinates of a Screw on a Cylindroid ....... 


37 


41. 


The Canonical Co- reciprocals 


38 


42. 


An Expression for the Virtual Coefficient 


38 


43. 


Equations of a Screw ........... 


38 


44. 


A Screw of Infinite Pitch 


39 


45. 


Indeterminate Screw ........... 


40 


46. 


A Screw at Infinity . . , . 


41 


47. 


Screws on One Axis . . . . . . . ... 


41 


48. 


Transformation of Screw Co-ordinates 


42 


49. 


Principal Screws on a Cylindroid 


43 




CHAPTER V. 






THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE. 




50. 


A Plane Representation 


45 


51. 


The Axis of Pitch 


46 


52. 


The Distance between Two Screws . . . 


47 



CONTENTS. IX 

PAGE 

53. The Angle between Two Screws 48 

54. The Triangle of Twists 49 

55. Decomposition of Twists and Wrenches ....... 50 

56. Composition of Twists and Wrenches 50 

57. Screw Co-ordinates 51 

58. Reciprocal Screws ....... .... 51 

59. Another Representation of the Pitch 52 

60. Pitches of Reciprocal Screws ......... 53 

61. The Virtual Coefficient 54 

62. Another Investigation of the Virtual Coefficient 55 

63. Application of Screw Co-ordinates 57 

64. Properties of the Virtual Coefficient 59 

65. Another Construction for the Pitch 59 

66. Screws of Zero Pitch 60 

67. A Special Case 60 

68. A Tangential Section of the Cylindroid 60 



CHAPTER VI. 
THE EQUILIBRIUM OF A RIGID BODY. 

69. A Screw System 62 

70. Constraints ............ 63 

71. Screw Reciprocal to a System 63 

72. The Reciprocal Screw System 63 

73. Equilibrium 64 

74. Reaction of Constraints 64 

75. Parameters of a Screw System 65 

76. Applications of Co-ordinates 65 

77. Remark on Systems of Linear Equations 67 



CHAPTER VII. 
THE PRINCIPAL SCREWS OF INERTIA. 

78. Introduction 69 

79. Screws of Reference 70 

80. Impulsive Screws and Instantaneous Screws 71 

81. Conjugate Screws of Inertia - . . . . 71 

82. The Determination of the Impulsive Screw 72 

83. System of Conjugate Screws of Inertia 72 

84. Principal Screws of Inertia . 73 

85. An Algebraical Lemma 75 

86. Another Investigation of the Principal Screws of Inertia .... 76 

87. Enumeration of Constants 78 

88. Kinetic Energy ........ 79 



X CONTENTS. 

PAGE 

89. Expression for Kinetic Energy 80 

90. Twist Velocity acquired by an Impulsive Wrench 81 

91. Kinetic Energy acquired by an Impulsive Wrench 82 

92. Formula for a Free Body 82 

93. Lemma ........ 83 

94. Euler s Theorem . 83 

95. Co-ordinates in a Screw System 83 

96. The Reduced Wrench 84 

97. Co-ordinates of Impulsive and Instantaneous Screws .... 85 



CHAPTER VIII. 
THE POTENTIAL. 

98. The Potential . . . . 87 

99. The Wrench evoked by Displacement 88 

100. Conjugate Screws of the Potential 89 

101. Principal Screws of the Potential 90 

102. Co-ordinates of the Wrench evoked by a Twist 91 

103. Form of the Potential 92 



CHAPTER IX. 
HARMONIC SCREWS. 

104. Definition of an Harmonic Screw . 94 

105. Equations of Motion ........ 96 

106. Discussion of the Results 99 

107. Remark on Harmonic Screws 100 



CHAPTER X. 
FREEDOM OF THE FIRST ORDER. 

108. Introduction 101 

109. Screw System of the First Order . . 101 

110. The Reciprocal Screw System 102 

111. Equilibrium 103 

112. Particular Case ..... 104 

113. Impulsive Forces 104 

114. Small Oscillations 105 

115. Property of Harmonic Screws 106 



CONTENTS. XI 



CHAPTER XI. 

FREEDOM OF THE SECOND ORDER. 
H 

oo 

116. The Screw System of the Second Order .... 

117. Applications of Screw Co-ordinates ..... 

118. Relation between Two Cylindroids 

119. Co-ordinates of Three Screws on a Cylindroid . 

120. Screws on One Line ........ 

121. Displacement of a Point 

122. Properties of the Pitch Conic 

123. Equilibrium of a Body with Freedom of the Second Order 

124. Particular Cases ..... ... 

125. The Impulsive Cylindroid and the Instantaneous Cylindroid 

126. Reaction of Constraints 

127. Principal Screws of Inertia ....... 

128. The Ellipse of Inertia . . . . . . 

129. The Ellipse of the Potential 

130. Harmonic Screws ......... 

131. Exceptional Case 

132. Reaction of Constraints. ..... 



CHAPTER XTI. 

PLANE REPRESENTATION OF DYNAMICAL PROBLEMS CONCERNING A BODY 
WITH Two DEGREES OF FREEDOM. 

133. The Kinetic Energy 120 

134. Body with Two Degrees of Freedom 120 

135. Conjugate Screws of Inertia 124 

136. Impulsive Screws and Instantaneous Screws 125 

137. Two Homographic Systems .......... 126 

138. The Homographic Axis 127 

139. Determination of the Homographic Axis . . . . . . . 128 

140. Construction for Instantaneous Screws . . . . . . 128 

141. Twist Velocity acquired by an Impulse ....... 129 

142. Another Construction for the Twist Velocity 129 

143. Twist Velocities on the Principal Screws 131 

144. Another Investigation of the Twist Velocity acquired by an Impulse . . 131 

145. A Special Case 133 

146. Another Construction for the Twist Velocity acquired by an Impulse . 134 

147. Constrained Motion 136 

148. Energy acquired by an Impulse 137 

149. Euler s Theorem 138 

150. To determine a Screw that will acquire a given Twist Velocity under 

a given Impulse 138 

151. Principal Screws of the Potential 140 



Xll CONTENTS. 

PAGE 

152. Work done by a Twist 141 

153. Law of Distribution of v 142 

154. Conjugate Screws of Potential 142 

155. Determination of the Wrench evoked by a Twist 143 

156. Harmonic Screws 143 

157. Small Oscillations in general 144 

158. Conclusion . 144 



CHAPTER XIII. 
THE GEOMETRY OF THE CYLINDROID. 

159. Another Investigation of the Cylindroid . 146 

160. Equation to Plane Section of Cylindroid 152 

161. Chord joining Two Screws of Equal Pitch 155 

162. Parabola . 157 

163. Chord joining Two Points 160 

164. Reciprocal Screws . . 181 

165. Application to the Plane Section 163 

166. The Central Section of the Cylindroid 166 

167. Section Parallel to the Nodal Line . 167 

168. Relation between Two Conjugate Screws of Inertia . . 168 



CHAPTER XIV. 
FREEDOM OF THE THIRD ORDER. 

169. Introduction 170 

170. Screw System of the Third Order 170 

171. The Reciprocal Screw System 171 

172. Distribution of the Screws 171 

173. The Pitch Quadric 172 

174. The Family of Quadrics 173 

175. Construction of a Three-system from Three given Screws . . . . 175 

176. Screws through a Given Point 176 

177. Locus of the feet of perpendiculars on the generators . . . . 178 

178. Screws of the Three-System parallel to a Plane 179 

179. Determination of a Cylindroid 180 

180. Miscellaneous Remarks 182 

181. Virtual Coefficients 183 

182. Four Screws of the Screw System 184 

183. Geometrical Notes 184 

184. Cartesian Equation of the Three-System 184 

185. Equilibrium of Four Forces applied to a Rigid Body .... 186 

186. The Ellipsoid of Inertia 187 

187. The Principal Screws of Inertia 188 

188. Lemma . . . 189 



CONTENTS. Xlll 

PAGE 

189. Relation between the Impulsive Screw and the Instantaneous Screw . 189 

190. Kinetic Energy acquired by an Impulse 189 

191. Reaction of the Constraints .......... 191 

192. Impulsive Screw is Indeterminate 191 

193. Quadric of the Potential 192 

194. The Principal Screws of the Potential . 192 

195. Wrench evoked by Displacement 193 

196. Harmonic Screws ............ 193 

197. Oscillations of a Rigid Body about a Fixed Point . . . . . . 194 



CHAPTER XV. 
THE PLANE REPRESENTATION OF FREEDOM OF THE THIRD ORDER. 

198. A Fundamental Consideration 197 

199. The Plane Representation 198 

200. The Cylindroid 199 

201. The Screws of the Three-System 200 

202. Imaginary Screws .......... 201 

203. Relation of the Four Planes to the Quadrics 202 

204. The Pitch Conies 204 

205. The Angle between Two Screws 204 

206. Screws at Right Angles 206 

207. Reciprocal Screws 206 

208. The Principal Screws of the System 207 

209. Expression for the Pitch 208 

210. Intersecting Screws in a Three-System 212 

211. Application to Dynamics 214 



CHAPTER XVI. 
FREEDOM OF THE FOURTH ORDER. 

212. Screw System of the Fourth Order 218 

213. Equilibrium with Freedom of the Fourth Order 219 

214. Screws of Stationary Pitch 221 

215. Applications of the Two-System . 224 

216. Application to the Three-System 226 

217. Principal Pitches of the Reciprocal Cylindroid 227 

218. Equations to the Screw in a Four-System 229 

219. Impulsive Screws and Instantaneous Screws . . . . . 229 

220. Principal Screws of Inertia in the Four-System . . . . . . 230 

221. Application of Euler s Theorem 231 

222. General Remarks 232 

223. Quadratic n-systems 233 

224. Properties of a Quadratic Two-System 234 

225. The Quadratic Systems of Higher Orders .... 235 



XIV CONTENTS. 

PAGE 

226. Polar Screws 238 

227. Dynamical Application of Polar Screws ....... 241 

228. On the Degrees of certain Surfaces . . . 242 



CHAPTER XVII. 
FREEDOM OF THE FIFTH ORDER. 

229. Screw Reciprocal to Five Screws 246 

230. Six Screws Reciprocal to One Screw ........ 247 

231. Four Screws of a Five-system on every Quadric 250 

232. Impulsive Screws and Instantaneous Screws ...... 251 

233. Analytical Method. 252 

234. Principal Screws of Inertia .......... 252 

235. The Limits of the Roots 253 

236. The Pectenoid 254 

CHAPTER XVIII. 
FREEDOM OF THE SIXTH ORDER. 

237. Introduction 258 

238. Impulsive Screws 258 

239. Theorem . 259 

240. Theorem 260 

241. Principal Axis 260 

242. Harmonic Screws 261 

CHAPTER XIX. 

HOMOGRAPHIC SCREW SYSTEMS. 

243. Introduction 262 

244. On Plane Homographic Systems . . . 262 

245. Homographic Screw Systems 263 

246. Relations among the Co-ordinates 263 

247. The Double Screws 264 

248. The Seven Pairs 264 

249. Homographic Ti-systems ........... 265 

250. Analogy to Anharmonic Ratio 266 

251. A Physical Correspondence .......... 267 

252. Impulsive and Instantaneous Systems 267 

253. Special type of Homography 268 

254. Reduction to a Canonical Form ......... 269 

255. Correspondence of a Screw and a System 270 

256. Correspondence of m and n Systems 271 

257. Screws common to the Two Systems 271 

258. Corresponding Screws denned by Equations 272 

259. Generalization of Anharmonic Ratio . . 273 



CONTENTS. XV 



CHAPTER XX. 
EMANANTS AND PITCH INVARIANTS. 

PAGE 

260. The Dyname 274 

261. Emanants 275 

262. Angle between Two Screws .......... 276 

263. Screws at Right Angles 276 

264. Conditions that Three Screws shall be parallel to a Plane . . . 277 

265. Screws on the same Axis 277 

266. A General Expression for the Virtual Coefficient 278 

267. Analogy to Orthogonal Transformation ....... 280 

268. Property of the Pitches of Six Co-reciprocals ...... 282 

269. Property of the Pitches of n Co-reciprocals ...... 285 

270. Theorem as to Signs 285 

271. Identical Formulae in a Co-reciprocal System 286 

272. Three Pitches Positive and Three Negative 287 

273. Linear Pitch Invariant Functions 287 

274. A Pitch Invariant 289 

275. Geometrical meaning ........... 290 

276. Screws at Infinity 291 

277. Expression for the Pitch 292 

278. A System of Emanants which are Pitch Invariants ... . 294 



CHAPTER XXI. 
DEVELOPMENTS OF THE DYNAMICAL THEORY. 

279. Expression for the Kinetic Energy 296 

280. Expression for the Twist Velocity 297 

281. Conditions to be fulfilled by Two Pairs of Impulsive and Instantaneous Screws 298 

282. Conjugate Screws of Inertia ........ 299 

283. A Fundamental Theorem ...... 300 

284. Case of a Constrained Rigid Body 303 

285. Another Proof 304 

286. Twist Velocity acquired by an Impulse 305 

287. System with Two Degrees of Freedom ...... 306 

288. A Geometrical Proof 30g 

289. Construction of Chiastic Homography on the Cylindroid . . . 307 

290. Homographic Systems on Two Cylindroids 307 

291. Case of Normal Cylindroids 30g 

292. General Conditions of Chiastic Homography .... 309 

293. Origin of the Formulae of 281 310 

294. Exception to be noted 312 

295. Impulsive and Instantaneous Cylindroids 312 

296. An Exceptional Case ...... 314 

297. Another Extreme Case 31g 

298. Three Pairs of Correspondents 31 7 



XVI CONTENTS. 

PAGE 

299. Cylindroid Reduced to a Plane 319 

300. A difficulty removed 320 

301. Two Geometrical Theorems . 320 



CHAPTER XXII. 

THE GEOMETRICAL THEORY. 

302. Preliminary 322 

303. One Pair of Impulsive and Instantaneous Screws 323 

304. An Important Exception 325 

305. Two Pairs of Impulsive and Instantaneous Screws 325 

306. A System of _ Rigid Bodies 326 

307. The Geometrical Theory of Three Pairs of Screws 330 

308. Another Method 332 

309. Unconstrained Motion in system of Second Order 332 

310. Analogous Problem in a Three-system 334 

311. Fundamental Problem with Free Body 336 

312. Freedom of the First or Second Order . 338 

313. Freedom of the Third Order. . . . 339 

314. General Case 339 

315. Freedom of the Fifth Order 340 

316. Principal Screws of Inertia of Constrained Body 341 

317. Third and Higher Systems 342 

318. Correlation of Two Systems of the Third Order 344 

319. A Property of Reciprocal Screw Systems 347 

320. Systems of the Fourth Order 348 

321. Systems of the Fifth Order 350 

322. Summary 350 

323. Two Rigid Bodies 351 

CHAPTER XXIII. 

VARIOUS EXERCISES. 

324. The Co-ordinates of a Rigid Body 355 

325. A Differential Equation satisfied by the Kinetic Energy .... 356 

326. Co-ordinates of Impulsive Screw in terms of the Instantaneous Screw . 356 

327. Another Proof of Article 303 357 

328. A more general Theorem 357 

329. Two Three-Systems 357 

330. Construction of Homographic Correspondents 358 

331. Geometrical Solution of the same Problem 359 

332. Co-reciprocal Correspondents in Two Three-systems 360 

333. Impulsive and Instantaneous Cylindroids 361 

334. The Double Correspondents on Two Cylindroids 363 

335. A Property of Co-reciprocals 364 

336. Instantaneous Screw of Zero Pitch 365 

337. Calculation of a Pitch Quadric 365 



CONTENTS. XVII 

CHAPTER XXIV. 

THE THEORY OF SCREW-CHAINS. 








PAGE 


338. 




367 


339. 




368 


340. 




368 


QzLl 




369 


Orr L. 

342. 


Freedom of the First Order 


369 


343. 


Freedom of the Second Order 


370 


344. 




374 






345. 




375 


346. 


Freedom of the Fourth Order 


377 


347. 


Freedom of the Fifth Order 


378 


348. 


Application of Parallel Projections ...... 


379 


349. 


Properties of this correspondence 


383 


350. 


Freedom of the Fifth Order 


384 


351. 


Freedom of the Sixth Order 


386 


352. 


Freedom of the Seventh Order 


386 


353. 


Freedom of the Eighth and Higher Orders ..... 


388 


354. 




388 


355. 


Twists on 6/^ + 1 Screw-chains ....... 


390 


356. 


Impulsive Screw-chains and Instantaneous Screw-chains 


392 


357. 


The principal Screw-chains of Inertia . .... . . 


394 


358. 


Conjugate Screw-chains of Inertia 


396 


359. 


Harmonic Screw-chains . 


397 



CHAPTER XXV. 
THE THEORY OF PERMANENT SCREWS. 

360. Introduction .... 399 

361. Different Properties of a Principal Axis ....... 400 

362. A Property of the Kinetic Energy of a System 401 

363. The Identical Equation in Screw-chain Co-ordinates 403 

364. The Converse Theorem 404 

365. Transformation of the Vanishing Emanant ....... 405 

366. The General Equations of Motion with Screw-chain Co-ordinates . . 405 

367. Generalization of the Eulerian Equations ....... 406 

368. The Restraining Wrench-chain 407 

369. Physical meaning of the Vanishing Emanant 408 

370. A Displacement without change of Energy ...... 408 

371. The Accelerating Screw-chain . 409 

372. Another Proof 409 

373. Accelerating Screw-chain and Instantaneous Screw-chain .... 410 

374. Permanent Screw-chains 410 

375. Conditions of a Permanent Screw-chain . . . . . . . 411 

376. Another identical Equation 412 



XVlll CONTENTS. 

PAGE 

377. Different Screws on the same Axis 414 

378. Co-ordinates of the Restraining Wrench for a Free Rigid Body . . 414 

379. Limitation to the position of the Restraining Screw .... 416 

380. A Verification 416 

381. A Particular Case 417 

382. Remark on the General Case 418 

383. Two Degrees of Freedom 419 

384. Calculation of T 420 

385. Another Method 420 

386. The Permanent Screw 421 

387. Geometrical Investigation 422 

388. Another Method 423 

389. Three Degrees of Freedom 426 

390. Geometrical Construction for the Permanent Screws 427 

391. Calculation of Permanent Screws in a Three-system 428 

392. Case of Two Degrees of Freedom 430 

393. Freedom of the Fourth Order . 431 

394. Freedom of the Fifth and Sixth Orders 432 

395. Summary ........... 432 



CHAPTER XXVI. 

AN INTRODUCTION TO THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 

396. Introduction 433 

397. Preliminary notions ........... 433 

398. The Intervene 434 

399. First Group of Axioms of the Content 435 

400. Determination of the Function expressing the Intervene between Two 

Objects on a Given Range 435 

401. Another Process ............ 441 

402. On the Infinite Objects in an Extent 442 

403. On the Periodic Term in the Complete Expression of the Intervene . 443 

404. Intervenes on Different Ranges in a Content ...... 444 

405. Another Investigation of the possibility of Equally Graduated Ranges . 446 

406. On the Infinite Objects in the Content ....... 447 

407. The Departure 448 

408. Second Group of Axioms of the Content 448 

409. The Form of the Departure Function 449 

410. On the Arrangement of the Infinite Ranges 449 

411. Relations between Departure and Intervene ...... 450 

412. The Eleventh Axiom of the Content 451 

413. Representation of Objects by Points in Space ...... 453 

414. Poles and Polars 454 

415. On the Homographic Transformation of the Content .... 454 

416. Deduction of the Equations of Transformation 455 

417. On the Character of a Homographic Transformation which Conserves 

Intervene . . 456 



CONTENTS. XIX 



PAGE 





418. The Geometrical Meaning of this Symmetric Function .... 461 

419. On the Intervene through which each Object is Conveyed . . . 464 

420. The Orthogonal Transformation 465 

421. Quadrics unaltered by the Orthogonal Transformation .... 466 

422. Proof that U and Q have Four Common Generators .... 467 

423. Verification of the Invariance of Intervene 468 

424. Application of the Theory of Emanants 469 

425. The Vector in Orthogonal Co-ordinates 470 

426. Parallel Vectors 472 

427. The Composition of Vectors 473 

428. Geometrical proof that Two Homonymous Vectors compound into One 

Homonymous Vector 475 

429. Geometrical proof of the Law of Permutability of Heteronymous Vectors 476 

430. Determination of the Two Heteronymous Vectors equivalent to any 

given Motor 476 

431. The Pitch of a Motor 478 

432. Property of Right and Left Vectors 478 

433. The^ Conception of Force in Non-Euclidian Space 480 

434. Neutrality of Heteronymous Vectors . . . - 480 

APPENDIX I. Notes on various points ....... 483 

II. A Dynamical Parable 496 

BIBLIOGRAPHICAL NOTES 510 

INDEX 540 



THE THEOEY OF SCKEWS. 



INTRODUCTION. 

THE Theory of Screws is founded upon two celebrated theorems. One 
relates to the displacement of a rigid body. The other relates to the forces 
which act on a rigid body. Various proofs of these theorems are well known 
to the mathematical student. The following method of considering them 
may be found a suitable introduction to the present volume. 

ON THE REDUCTION OF THE DISPLACEMENT OF A RIGID BODY TO ITS 

SIMPLEST FORM. 

Two positions of a rigid body being given, there is 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. He has shown that a 
free rigid body can be moved from any one specified position to any other 
specified position by a movement consisting of a rotation around a straight 
line accompanied by a translation parallel to the straight line. 

Regarding the rigid body as an aggregation of points its change of 
place amounts to a transference of each point P to a new point Q. The 
initial and the final positions of the body being given each point P corre 
sponds to one Q, and each Q to one P. If the coordinates of P be given 
then those of Q will be determined, and vice versa. If we represent P by 
its quadriplanar coordinates x l , ae z , x 3 , x 4> then the quadriplanar coordinates 
2/i. 2/2> 2/s- 2/4 f Q must be uniquely determined. There must, therefore, 
be equations connecting these coordinates, and as the correspondence is 
essentially of the one-to-one type these equations must be linear. We 
shall, therefore, write them in the form 

y, = (11) x, + (12) x z + (13) x. + (14) x,, 
y* = (21) x, + (22) x, + (23) x. A + (24) x t , 
y 9 = (31) x, + (32) x, + (33) x a + (34) a? 4 , 
y 4 = (41) ^ + (42) x z + (43) x, + (44) x.. 
B. 1 



2 INTRODUCTION. 



If we make y l = px 1} y 2 = p# 2 , 2/s = P x a> 2/4 = P x * we can eliminate x l ,x z , x a , 
\ and obtain the following biquadratic for p, 



(11) -p, 


(12) 


(13) 


(14) 


(21) 


(22) -p, 


(23) 


(24) 


(31) 


(32) 


(33) -p, 


(34) 


(41) 


(42) 


(43) 


(44) - p 



= 0. 



The four roots of this indicate four double points, i.e. points which remain 
unaltered. But these points are not necessarily distinct or real. 

What we have written down is of course the general homographic trans 
formation of the points in space. For the displacement of a rigid system is 
a homographic transformation of all its points, but it is a very special kind of 
homographic transformation, as will be made apparent when we consider what 
has befallen the four double points. 

In the first place, since the distance between every two points before the 
transformation is the same as their distance after the transformation it 
follows that every point in the plane at infinity before any finite trans 
formation must be in the same plane afterwards. Hence the plane at infinity 
remains in the same position. Further, a sphere before this transformation is 
still a sphere after it. But it is well known that all spheres intersect the 
plane at infinity in the same imaginary circle H. Hence we see not only that 
the plane at infinity must remain unaltered by the transformation but that a 
certain imaginary circle in that plane is also unaltered. 

A system of points P lt P 2 , P 3 , &c. on this circle fl will, therefore, have 
as their correspondent points Q,, Q.,, Q 3 , <foc. also on fl As all anhar- 
monic ratios are unaltered by a linear transformation it follows that the 
systems P lt P 2 , P 3 , &c. and Q l} Q 2 , Q :t , &c. are homographic. There will, 
therefore, be two double points of this homography, O l and 0, and these will 
be the same after the transformation as they were before. They are, there 
fore, two of the four double points of which we were in search. 

It should be remarked that the points Oj and 0., cannot coincide, for if 
they coincide in 0, then must be the double point corresponding to a 
repeated root of the biquadratic for p. But such a root is real. Hence 
must be real. But every point on fl is imaginary. Hence this case is 
impossible. 

As Q is unaltered and Oi and 2 are fixed, the tangents at 0, and 2 
are fixed, and so is therefore T, the intersection of these tangents; this is 
accordingly the third of the four points wanted. It lies in the plane at 
infinity, but is a real point. The ray O^Og, is also real ; it is the vanishing 
line of the planes perpendicular to the parallel rays, of which T is the 
vanishing point. 



INTRODUCTION. 3 

In general in any homographic transformation there cannot be four distinct 
double points in a plane, unless every point of the plane is a double point. 
For suppose P 1; P 2 , P 3 , P 4 were four distinct coplanar double points and that 
any other point R had a correspondent R 1 . Draw the conic through P lt P 2 , 
P 3 , P 4 , R. Then R must lie on this conic because the anharmonic ratios 
R(P lt P 2 , P 3 , P 4 ) and R (l\, P 2 , P 3 , P 4 ) are equal. We have also 
P,(P 2 , P 3 , P 4) R) and P, (P 2 , P 3 , P 4 , R ) equal, but this is impossible if R 
and R be distinct. R is therefore a double point. 

In the case of the displaced rigid body suppose there is a fourth distinct 
double point in the plane at infinity. Each ray connected with the body 
will then have one double point at infinity, so that after the transformation 
the ray must again pass through the same point, i.e. the transformed position 
of each ray must be parallel to its original position. This is a special form of 
displacement. It is merely a translation of the whole rigid system in which 
every ray moves parallel to itself. 

In the more general type of displacement there can therefore be no 
double point distinct from T, 0], 2 and lying in the plane at infinity. Nor 
can there be in general another double point at a finite position T . For if 
so, then the ray TT is unaltered in position, and any finite point T" on the 
ray TT will be also unaltered, since this homographic transformation does 
not alter distances. Hence every point on TT is a double point. Here 
again we must have fallen on a special case where the double points instead 
of being only four have become infinitely numerous. In this case every point 
on a particular ray has become a double point. The change of the body from 
one position to the other could therefore be effected by simple rotation around 
this ray. 

There must however be four double points even in the most general case. 
Not one of these is to be finite, and in the plane at infinity not more than 
three are to be distinct. The fourth double point must be in the plane at 
infinity, and there it must coincide with either 0,, 2 or T. Thus we learn 
that the most general displacement of a rigid system is a homographic trans 
formation of all its points with the condition that two of its double points are 
on the imaginary circle fi in the plane at infinity, while the pole of their chord 
gives a third. Of these three one, we shall presently see which one, is to be 
regarded as formed of two coincident double points. 

All rays through T are parallel rays, and hence we learn that in the 
general displacement of a rigid body there is one real parallel system of 
rays each of which L is transformed into a parallel ray L . Let A be any 
plane perpendicular to this parallel system. Let L and L cut A in the points 
R and R. Then as L and L move, R and R are corresponding points in two 
plane homographic systems. Any two such systems in a plane will of course 

1-2 



4 INTRODUCTION. 

have three double points. The special feature of this homographic trans 
formation is that every circle is transformed into a circle. Each circle passes 
through the two circular points at infinity in its plane. These two points in 
A are therefore two of the double points of the plane homographic trans 
formation. There remains one real point X in A which is common to the 
two systems. The normal S to A drawn through X is therefore the one and 
only ray which the homographic transformation does not alter. 

This shows that in the most general change of a rigid system from one 
position to another there is one real ray unaltered Hence every point on S 
before the transformation is also on S afterwards. There must therefore be 
two double points distinct or coincident on S. But we have already proved 
that in the general case there is no finite double point. Hence S must 
have two coincident double points at T. Thus we learn that in the general 
transformation of a system which is equivalent to the displacement of a rigid 
body, there is one real point at infinity which is the result of two coinciding 
double points, and the polar of this point with respect to the imaginary 
circle on the plane at infinity cuts that circle in the two other double points. 

The displacement of the rigid body can thus be produced either by 
rotating the body around S or by translating the body parallel to 8, or by 
a combination of such movements. We are therefore led to the funda 
mental theorem discovered by Chasles. 

Any given displacement of a rigid body can be effected by a rotation about 
an axis combined with a translation parallel to that axis. 

Of much importance is the fact that this method of procedure is in 
general unique. It is easily seen that there is only one axis by rotation about 
which, and translation parallel to which, the body can be brought from one 
given position to another given position. Suppose there were two axes P 
and Q, which possessed this property, then by the movement about P, all the 
points of the body originally on the line P continue thereon ; but it cannot 
be true for any other line that all the points of the body originally on that 
line continue thereon after the displacement. Yet this would have to be true 
for Q, if by rotation around Q and translation parallel thereto, the desired 
change could be effected. We thus see that the displacement of a rigid body 
can be made to assume an extremely simple form, in which no arbitrary 
element is involved. 



ON THE REDUCTION OF A SYSTEM OF FORCES APPLIED TO A RIGID BODY 

TO ITS SIMPLEST FORM. 

It has been discovered by Poinsot that any system of forces which act 
upon a rigid body can be replaced by a single force, and a couple in a plane 



INTRODUCTION. 5 

perpendicular to the force. Thus a force, and a couple in a plane perpen 
dicular to the force, constitute an adequate representation of any 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 compounded into a force acting at the 
origin, and a couple. Wherever the origin be taken, the magnitude and 
direction of the force are both manifestly invariable ; but this is not the case 
either with the moment of the couple or the direction of its axis. 

The origin, however, can always be 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 com 
ponent 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 that force. 

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. Sup 
pose two lines possessed the property, then if the force and couple belonging 
to one were reversed, they must destroy the force and couple belonging 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 
acting along these must be equal and opposite. The two forces would there 
fore form a couple in a plane perpendicular to that of the couple which is 
found by compounding 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 simple form, in which no arbitrary element is 
involved. 

These two principles being established we are able to commence the 
Theory of Screws. 



CHAPTER I. 



TWISTS AND WRENCHES. 

1. Definition of the word Pitch. 

The direct problem offered by the Dynamics of a Rigid Body may be 
thus stated. To determine at any instant the position of a rigid body 
subjected to certain constraints and acted upon by certain forces. We may 
first inquire as to the manner in which the solution of this problem ought to 
be presented. Adopting one position of the body as a standard of reference, 
a complete solution of the problem ought to provide the means of deriving 
the position at any epoch from the standard position. We are thus led to 
inquire into the most convenient method of specifying one position of a body 
with respect to another. 

To make our course plain let us consider the case of a mathematical point. 
To define the position of the point P with reference to a standard point A, 
there can be no more simple method than to indicate the straight line along 
which it would be necessary for a particle to travel from A in order to arrive 
at P, as well as the length of the journey. There is a more general 
method of defining the position of a rigid body with reference to a certain 
standard position. We can have a movement prescribed by which the body 
can be brought from the standard position to the sought position. It was 
shown in the Introduction that there is one simple movement which will 
always answer. A certain axis can be found, such that if the body be rotated 
around this axis through a determinate angle, and translated parallel to the 
axis for a determinate distance, the desired movement will be effected. 

It will simplify the conception of the movement to suppose, that at each 
epoch of the time occupied in the operations producing the change of position, 
the angle of rotation bears to the final angle of rotation, the same ratio 
which the corresponding translation bears to the final translation. Under 
these circumstances the motion of the body is precisely the same as if it were 
attached to the nut of a uniform screw (in the ordinary sense of the word), 



1-3] TWISTS AND WRENCHES. 7 

which had an appropriate position in space, and an appropriate number of 
threads to the inch. 

In the Theory of Screws the word pitch is employed in a particular sense 
that must be carefully noted. We define the pitch of a screw to be the 
rectilinear distance through which the nut is translated parallel to the axis 
of the screw, while the nut is rotated through the angular unit of circular 
measure. The pitch is thus a linear magnitude. It follows from this 
definition that the rectilinear distance parallel to the axis of the screw 
through which the nut moves when rotated through a given angle is simply 
the product of the pitch of the screw and the circular measure of the 
angle. 

2. Definition of the word Screw. 

It is a fundamental principle of the theory developed in these pages 
that the dynamical significance of screws is precisely analogous to their 
kinematical significance. It is, therefore, essential that in the formal 
definition of the particular sense the word screw is to bear in this volume 
no prominence can be assigned to kinematical terms or conceptions unless 
it can be equally given to dynamical terms and conceptions. This condition 
is fulfilled by excluding both Kinematics and Dynamics and constituting the 
screw as the geometrical entity thus described. 

A screw is a straight line with which a definite linear magnitude termed the 
pitch is associated. 

We shall often denote a screw by a symbol, and then usually by a small 
Greek letter. With reference to these symbols, a caution may be necessary. 
If, for example, a screw be denoted by a, then a is not an ordinary algebraic 
quantity. It is a symbol which denotes all that is included in the conception 
of a screw, and requires five quantities for its specification ; of these four are 
required to determine the position of the straight line, and the pitch must be 
specified by a fifth. It will often be convenient to denote the pitch by a 
symbol, derived from the symbol employed to denote the screw to which the 
pitch belongs. The pitch of a screw is accordingly represented by appending 
to the letter p a suffix denoting the screw. For example, p a denotes the pitch 
of a and is an ordinary algebraical quantity. 

3. Definition of the word Twist. 

We have next to define the use to be made of the word tiuist. 

A body is said to receive a twist about a screw when it is rotated uniformly 
about the screw, while it is translated uniformly parallel to the screw, through 
a distance equal to the product of the pitch and the circular measure of the 
angle of rotation. 



8 THE THEORY OF SCREWS. [4, 

4. A Geometrical Investigation. 

We can now demonstrate that whenever a body admits of an indefinitely 
small movement of a continuous nature it must be capable of executing that 
particular kind of movement denoted by a twist about a screw. 

Let AI be a standard position of the body, and let P be any marked 
point of the body initially at P lt As the body is displaced continuously to a 
neighbouring position, P will generally pursue a certain trajectory which, as 
the motion is small, may be identified with its tangent on which P n is a 
point adjacent to P,. In travelling from P, to P n , P passes through the 
several positions, P 2 , . . . P^. In a similar manner every other point, Q 1} of the 
rigid body will pass through a series of positions, Q 2 , &c., to Q n . We thus 
have the points of the body initially at P 1} Q lt R 1} respectively, and each 
moves along a straight line through the successive systems of positions 
P*, Qa, RZ, &c., on to the final position P n , Q n , R n . We may thus think of 
the consecutive positions occupied by the body A 2> A 3 , &c., as defined by the 
groups of points P lt Q 1} R 1 and P 2 , Q 2 , R 2 , &c. We have now to show that if 
the body be twisted by a continuous screw motion direct from A 1 to A n , it 
will pass through the series of positions A 2> A 3 , &c. It must be remembered 
that this is hardly to be regarded as an obvious consequence. From the 
initial position A 1 to the final position A n , the number of routes are generally 
infinitely various, but when these situations are contiguous, it is always 
possible to pass by a twist about a screw from A 1 to A n via the positions 
A 2 , A 3 . .. A n _!. 

Suppose the body be carried direct by a twist about a screw from the 
position AT, to the position A n . Since this motion is infinitely small, each 
point of the body will be carried along a straight line, and as Pj is to be 
conveyed to P B , this straight line can be no other than the line PiP n . 
In its progress P l will have reached the position P 2 , and when it is 
there the points Q l} R t will each have advanced to certain positions along 
the lines QtQ n and R^n, respectively. But the points reached by Q l and 
R 1 can be no other than the points Q. 2 and R 2 , respectively. To prove this 
we shall take the case where P 1} Q 1} R 1 are collinear. Suppose that when 
P! has advanced to P 2 , Q 1 shall not have reached Q.,, but shall be at 
the intermediate point Q . (Fig. 1.) Then the line P& will have moved to 
P 2 Q , and as 7^ can only be conveyed along R^.,, while at the same time 
it must lie along P 2 Q , it follows that the lines P 2 Q and R^ 2 must intersect 
at the point R , and consequently all the lines in this figure lie in a plane. 
Further, P 2 Q 2 and P 2 Q are each equal to PjQi, as the body is rigid, and so 
also P 2 R and P 2 R 2 are equal to P^. Hence it follows that QiQ and R^o 
are parallel, and consequently all the points on the line P&Ri are displaced 
in parallel directions. It would hence follow that the motion of every point 
in the body was in a parallel direction, and that consequently the entire 



5] TWISTS AND WRENCHES. 9 

movement was simply a translation. But even in this case it would be 
impossible for the points Q and R to be distinct from Q. 2 and R 2 , because, 




Fig. l. 

when a body is translated so that all its points move in parallel lines, it is 
impossible, if the body be rigid, for the distances traversed by each point not 
to be all equal. We have thus demonstrated that if a body is free to 
move from a position A 1 to an adjacent position A n by an infinitely small 
but continuous movement, it is also free to move through the series of 
positions A. 2> A 3 , &c., by which it would be conveyed from A^ to A n by a twist. 

We may also state the matter in a somewhat different manner, as 
follows : It would be impossible to devise a system of constraints which 
would permit a body to be moved continuously from A 1 to A n , and would at the 
same time prohibit the body from twisting about the screw which directly 
conducts from A l to A n . Of course this would not be true except in the case 
where the motion is infinitely small. The connexion of this result with the 
present investigation is now obvious. When A is the standard position of the 
body, and B an adjacent position into which it can be moved, then the body 
is free to twist about the screw defined by A and B. 

5. The canonical form of a small displacement. 

In the Theory of Screws we are only concerned with the small displace 
ments of a system, and hence we can lay down the following fundamental 
statement. 

The canonical form to which the displacement of a rigid body can be 
reduced is a twist about a screw. 

If a body receive several twists in succession, then the position finally 
attained could have been reached in a single twist, which is called the 
resultant tivist. 



10 THE THEORY OF SCREWS. [5- 

Although we have described the twist as a compound movement, yet in 
the present method of studying mechanics it is essential to consider the 
twist as one homogeneous quantity. Nor is there anything unnatural in 
such a supposition. Everyone will admit that the relation between two 
positions of a point is most simply presented by associating the purely 
metric element of length with the purely geometrical conception of a 
directed straight line. In like manner the relation between two positions of 
a rigid body can be most simply presented by associating a purely metric 
element with the purely geometrical conception of a screw, which is merely a 
straight line, with direction, situation, and pitch. 

It thus appears that a twist bears the same relation to a rigid body which 
the ordinary vector bears to a point. Each just expresses what is necessary 
to express the transference of the corresponding object from one given position 
to another*. 

6. Instantaneous Screws. 

Whatever be the movement of a rigid body, it is at every instant twisting 
about a screw. For the movement of the body when passing from one 
position to another position indefinitely adjacent, is indistinguishable from 
the twist about an appropriately chosen screw by which the same displacement 
could be effected. The screw about which the body is twisting at any 
instant is termed the instantaneous screw. 

7. Definition of the word Wrench. 

It has been explained in the Introduction that a system of forces 
acting upon a rigid body may be generally expressed by a certain force 
and a couple whose plane is perpendicular to the force. We now employ 
the word wrench, to denote a force and a couple in a plane perpendicular to 
the force. The quotient obtained by dividing the moment of the couple by 
the force is a linear magnitude. Everything, therefore, which could be 
specified about a wrench is determined (if the force be given in magnitude), 
when the position of a straight line is assigned as the direction of the force, 
and a linear magnitude is assigned as the quotient just referred to. 
Remembering the definition of a screw ( 2), we may use the phrase, 
ivrench on a screw, meaning thereby, a force directed along the screw and 
a couple in a plane perpendicular to the screw, the moment of the couple 
being equal to the product of the force and the pitch of the screw. Hence 
we may state that 

The canonical form to which a system of forces acting on a rigid body 
can be reduced is a wrench on a screw. 

* Compare M. Rene de Saussure, American Journal of Mathematics, Vol. xvm. No. 4, p. 337. 



TWISTS AND WRENCHES. 



11 



If a rigid body be acted upon by several wrenches, then these wrenches 
could be replaced by one wrench which is called the resultant wrench. 

A twist about a screw a requires six algebraic quantities for its complete 
specification, and of these, five are required to specify the screw a. The sixth 
quantity, which is called the AMPLITUDE OF THE TWIST, and is denoted by 
a , expresses the angle of that rotation which, when united with a translation, 
constitutes the entire twist. 

The distance of the translation is the product of the amplitude of the twist 
and the pitch of the screw, or in symbols, a p a . The sign of the pitch 
expresses the sense of the translation corresponding to a given rotation. 

If the pitch be zero, the twist reduces to a pure rotation around a. If 
the pitch be infinite, then a finite twist is not possible except the amplitude 
be zero, in which case the twist reduces to a pure translation parallel to a. 

A wrench on a screw a requires six algebraic quantities for its complete 
specification, and of these, five are required to specify the screw a. The sixth 
quantity, which is called the INTENSITY OF THE WRENCH, and is denoted by 
a", expresses the magnitude of that force which, when united with a couple, 
constitutes the entire wrench. 

The moment of the couple is the product of the intensity of the wrench 
and the pitch of the screw, or in symbols, a" p a . The sign of the pitch 
expresses the direction of the moment corresponding to a given force. 

If the pitch be zero, the wrench reduces to a pure force along a. If the 
pitch be infinite, then a finite wrench is not possible except the intensity be 
zero, in which case the wrench reduces to a couple in a plane perpendicular 
to a. 

In the case of a twisting motion about a screw a. the rate at which the 
amplitude of the twist changes is called the TWIST VELOCITY arid is denoted 
by a. 

8. Restrictions. 

It is first necessary to point out the restrictions which we shall impose 
upon the forces. The rigid body M, whose motion we are considering, is 
presumed to be acted upon by the same forces whenever it occupies the same 
position. The forces which we shall assume are to be such as form what is 
known as a conservative system. Forces such as those due to a resisting medium 
are excluded, because such forces do not depend merely on the position of 
the body, but on the manner in which the body is moving through that 
position. The same consideration excludes friction which depends on the 
direction in which the body is moving through the position under considera 
tion. 



12 THE THEORY OF SCREWS. [8 

But the condition that the forces shall be defined, when the position is 
given, is still not sufficiently precise. We might include, in this restricted 
group, forces which could have no existence in nature. We shall, therefore, 
add the condition that the system is to be one in which the continual creation 
of energy is impossible. 

An important consequence of this restriction is stated as follows : The 
quantity of energy necessary to compel the body M to move from the 
position A to the position B, is independent of the route by which the change 
has been effected. 

Let L and M be two such routes, and suppose that less energy was 
required to make the change from A to B via L than via M. Make the 
change via L, with the expenditure of a certain quantity of energy, and then 
allow the body to return via M. Now, since at every stage of the route M 
the forces acting on the body are the same whichever way the body be 
moving, it follows, that in returning from B to A via M, the forces will give 
out exactly as much energy as would have been required to compel the body 
to move from A to B via M ; but by hypothesis this exceeds the energy 
necessary to make the change via L, and hence, on the return of the body to 
A, there is a clear gain of a quantity of energy, while the position of the body 
and the forces are the same as at first. By successive repetitions of the 
process an indefinite quantity of energy could be created from nothing. This 
being contrary to experience, compels us to admit that the quantity of energy 
necessary to force the body from A to B is independent of the route 
followed. 

It follows that the amount of work done in a number of twists against 
a wrench is equal to the work that would be done in the resultant twist. 

For, the work done in producing a given change of position is independent 
of the route. 

We may calculate the work done in a twist against a wrench by deter 
mining the amount of work done against three forces which are equivalent 
to the wrench, in consequence of the movements of their points of application 
which are caused by the twist. 

We shall assume the two lemmas 1st. The work done in the displace 
ment of a rigid body against a force is the same at whatever point in its line 
of application the force acts. 2nd. The work done in the displacement of a 
point against a number of forces acting at that point, is equal to the work 
done in the same displacement against the resultant force. 

The theorem to be proved is as follows : The amount of work done in a 
given twist against a number of wrenches, is equal to the work done in the 
same twist against the resultant wrench. 



8] TWISTS AND WRENCHES. 13 

Let n wrenches, which consist of 3w forces acting at A lt ... A 3n , compound 
into one wrench, of which the three forces act at P, Q, R. The force at A k 
may generally be decomposed into three forces along PA^, QA/c, RA^. By 
the 2nd lemma the amount of work ( W) done against the 3w original forces, 
equals the amount of work done against the 9w components. It, therefore, 
appears from the 1st lemma, that W will still be the amount of work done 
against the 9n components, of which 3/i act at P, 3w at Q, 3n at R. Finally, 
by the 2nd lemma, W will also be the amount of work done by the original 
twist against the three resultants formed by compounding each group at 
P, Q, R. But these resultants constitute the resultant wrench, whence 
the theorem has been proved. 

We thus obtain the following theorem, which we shall find of great 
service throughout this book. 

If a series of twists A l ,...A m , would compound into one twist A, and 
a series of wrenches B 1 ,...B n , would compound into one wrench B, then 
the energy that would be expended or gained when the rigid body per 
forms the twist A, under the influence of the wrench B, is equal to the 
algebraic sum of the mn quantities of energy that would be expended or 
gained when the body performs severally each twist A lt ...A m under the 
influence of each wrench B l ,...B n . 

We have now explained the conceptions, and the language in which 
the solution of any problem in the Dynamics of a rigid body may be pre 
sented. A complete solution of such a problem must provide us, at each 
epoch, with a screw, by a twist about which of an amplitude also to be 
specified, the body can be brought from a standard position to the position 
occupied at the epoch in question. It will also be of much interest to know 
the instantaneous screw about which the body is twisting at each epoch, as 
well as its twist velocity. Nor can we regard the solution as quite complete, 
unless we also have a clear conception of the screw on which all the forces 
acting on the body constitute a wrench of which we should also know the 
intensity. 

There is one special feature which characterises that portion of Dynamics 
which is discussed in the present treatise. We shall impose no restrictions 
on the form of the rigid body, and but little 011 the character of the con 
straints by which its movements are limited, or on the forces to which the 
rigid body is submitted. The restriction which we do make is that the body, 
while the object of examination, remains in, or indefinitely adjacent to, its 
original position. 

As a consequence of this restriction, we here make the remark that the 
amplitude of a twist is henceforth to be regarded as a small quantity. 



14 THE THEORY OF SCREWS. [8 

If it be objected, that with so great a restriction as that just referred to, 
only a limited field of inquiry remains, the answer is as follows : A perfectly 
general investigation could yield but a slender harvest of interesting or 
valuable results. All the problems of Physical importance are special cases 
of the general question. Thus, a special character in the constraints has pro 
duced the celebrated problem of the rotation of a rigid body about a fixed 
point. To vindicate our particular restriction it seems only necessary to 
remark, that the restricted inquiry still includes the theory of Equilibrium, 
of Impulsive Forces, and of Small Oscillations. 

Whatever novelty may be found in the following pages will, it is 
believed, be largely due to the circumstance that, with the important 
exception referred to, all the conditions of each problem are of absolute 
generality. 



CHAPTER II. 



THE CYLINDROID. 



9. Introduction. 



Let a and /3 be any two screws which we shall suppose to be fixed both 
in position and in pitch. Let a body receive a twist of amplitude of about 
a, followed by a twist of amplitude /3 about /3. The position attained 
could have been arrived at by a single twist about some third screw p with 
an amplitude p. We are always to remember that the amplitudes of the 
twists are infinitely small quantities. With this assumption the order in 
which the twists about a and /3 are imparted will be immaterial in so far 
as the resulting displacements are concerned. The position attained is the 
same whether a follows /3 or /3 follows a . 

Any change in a or in /3 will of course generally entail a change both in 
the pitch and in the position of p. It might thus seem that p depended 
upon two parameters, and that consequently the different positions of p 
would form a doubly infinite series, known in linear geometry as a 
congruence. But this is not the case, for we prove that p depends only 
upon the ratio of a to /3 and is thus only singly infinite. 

Take any point P and let h a be the perpendicular distance from P to a, 
while p a is as usual the pitch of the screw; then the point P is transferred 
by the twist about a through the distance 

V^a 2 + / a 2 a . 
The twist about /3 conveys P to a distance 



The resultant of these two displacements conveys P in a direction which 
depends upon the ratio of a to /3 , and not upon their absolute magnitudes. 

Let P and Q be two points on p, then the resultant displacement will 
convey P and Q to points P and Q respectively which are also on the axis 



16 THE THEORY OF SCREWS. [9, 

of p. Suppose that a and ft be varied while their ratio is preserved P 
and Q will then be transferred to P" and Q" while by the property just 
proved P, P , P" will be collinear and so will Q, Q , Q". It therefore follows 
that as P, P , Q, Q are collinear so will P, Q, P", Q" be collinear. The line 
PQ will therefore be displaced upon itself for every pair of values a and 
$ which retain the same ratio. The position of the resultant screw is thus 
not altered by any changes of a! and ft , which preserves their ratio. 

Let f be the angle between a and ft. We take the case of a point P 
at an infinite distance on the common perpendicular to a and (3. This 
point is displaced through a distance equal to 



h \fa!- + ft 2 + 2a /3 cos &>, 

where h stands for the infinite perpendicular distance from P to a or to ft. 
This displacement of P is normal to p which itself intersects at right angles 
the common perpendicular to a and ft. As the perpendicular distance from 
P to p can only differ by a finite quantity from h 



hp = h Va 2 + /3 2 + 2a ft cos a>, 
or 

p = Va 7 - + ft - 2 +2a ft cos^a. 

This determines the amplitude of the resulting twist which is, it may be 
noted, independent of the pitches. 

Let <j) be the angle between the directions in which a point Q on p is 
displaced by the twists about a and ft, then the square of the displacement 
of Q will be 

(pa 2 + h a a ) a 2 + (pi + hi) ft 2 + 2 Vp a 2 + h* \?pi + hi aft cos cf> ; 
but this may also be written 



whence we see that p p depends only on the ratio of a to ft . 

The pitch and the position of p thus depend on the single numerical 
parameter expressing the ratio of a and ft . As this parameter varies so 
will p vary, and it must in successive positions coincide with the several 
generators of a certain ruled surface. Two of these generators will be the 
situations of a and of ft corresponding to the extreme values of zero and 
infinity respectively, which in the progress of its variation the parameter 
will assume. 

We shall next ascertain the laws according to which twists (and wrenches) 
must be compounded together, that is to say, we shall determine the single 
screw, one twist (or wrench) about which will produce the same effect on the 



10] THE CYLINDROID. 17 

body as two or more given twists (or wrenches) about two or more given 
screws. It will be found to be a fundamental point of the present theory 
that the rules for the composition of twists and of wrenches are identical*. 

10. The Virtual Coefficient. 

Suppose a rigid body be acted upon by a wrench on a screw ft, of which the 
intensity is ft". Let the body receive a twist of small amplitude a around a 
screw a. It is proposed to find an expression for the energy required to effect 
the displacement. 

Let d be the shortest distance between a and ft, and let 6 be the angle 
between a and ft. Take a as the axis of x, the common perpendicular to a. 
and ft as the axis of z, and a line perpendicular to x and z for y. If we 
resolve the wrench on ft into forces X, Y. Z, parallel to the axes, and couples 
of moments L, M, N, in planes perpendicular to the axes we shall have 

X=ft"cosO; Y=ft"smO; Z=0; 
L = p pfi cos - ft"d sin ; M = ft" Pii sin + ft"d cos ; 

N = 0. 

We thus replace the given wrench by four wrenches, viz., two forces and 
two couples, and we replace the given twist by two twists, viz., one rotation 
and one translation. The work done by the given twist against the given 
wrench must equal the sum of the eight quantities of work done by each of 
the two component twists against each of the four component wrenches. 
Six of these quantities are zero. In fact a rotation through the angle a 
around the axis of x can do work only against L, the amount being 
aft" (p ft cos - d sin 0). 

The translation p a a parallel to the axis of x can do work only against 
X, the amount being 

a ft"p a cos 0. 

Thus the total quantity of work done is 

aft" {(pa+pp) cos d sin 0}. 
The expression 

i [(P + PP) cos d sin 0] 
is of great importance in the present theoryf. It is called the virtual 

* That the analogy between the composition of forces and of rotations can be deduced from 
the general principle of virtual velocities has been proved by Rodrigues (Liouville s Journal, t. 5, 
1840, p. 436). 

+ The theory of screws has many points of connexion with certain geometrical researches on the 
linear complex, by Pliicker and Klein. Thus the latter has shown (Mathematische Annalen, Band 
n., p. 368 (1869)), that if p &nd p s be each the "Hauptparameter" of a linear complex, and if 

(P a +Pft) cos O-d sin = 0, 

where d and relate to the principal axes of the complexes, then the two complexes possess a 
special relation and are said to be in " involution." 

B. 2 



18 THE THEORY OF SCREWS. [10- 

coefficient of the two screws a and ft, and may be denoted by the 
symbol 



11. Symmetry of the Virtual Coefficient. 

An obvious property of the virtual coefficient is of great importance. If 
the two screws a and ft be interchanged, the virtual coefficient remains 
unaltered. The identity of the laws of composition of twists and wrenches 
can be deduced from this circumstance*, and also the Theory of Reciprocal 
Screws which will be developed in Chap. III. 

12. Composition of Twists and Wrenches. 

Suppose three twists about three screws a, ft, 7, possess the property 
that the body after the last twist has the same position which it had before 
the first : then the amplitudes of the twists, as well as the geometrical rela 
tions of the screws, must satisfy certain conditions. The particular nature 
of these conditions does not concern us at present, although it will be fully 
developed hereafter. 

We may at all events conceive the following method of ascertaining these 
conditions : 

Since the three twists neutralize it follows that the total energy ex 
pended in making those twists against a wrench, on any screw 77, must be 
zero, whence 

a trar, + ft ^ftr, + J^yr, = 0. 

This equation is one of an indefinite number (of which six can be shown 
to be independent) obtained by choosing different screws for 77. From 
each group of three equations the amplitudes can be eliminated, and four of 
the equations thus obtained will involve all the purely geometrical conditions 
as to direction, situation, and pitch, which must be fulfilled by the screws 
when three twists can neutralize each other. 

But now suppose that three wrenches equilibrate on the three screws 
a, ft, 7. Then the total energy expended in a twist about any screw 77 against 
the three wrenches must be zero, whence 

<*"^a,, + ft ^fr + j ^yr, = 0. 

An indefinite number of similar equations, one in fact for every screw 77, must 
be also satisfied. 

By comparing this system of equations with that previously obtained, it 
is obvious that the geometrical conditions imposed on the screws a, ft, 7, in 

* This pregnant remark, or what is equivalent thereto, is due to Klein (Math. Ann., Vol. iv. 
p. 413 (1871)). 



13] THE CYLINDROID. 19 

the two cases are identical. The amplitudes of the three twists which 
neutralise are, therefore, proportional to the intensities of the three wrenches 
which equilibrate. 

When three twists (or wrenches) neutralise, then a twist (or wrench) 
equal and opposite to one of them must be the resultant of the other two. 
Hence it follows that the laws for the composition of twists and of wrenches 
must be identical. 

13. The Cylindroid. 

We next proceed to study the composition of twists and wrenches, and 
we select twists for this purpose, though wrenches would have been equally 
convenient. 

A body receives twists about three screws ; under what conditions will 
the body, after the last twist, resume the same position which it had before 
the first ? 

The problem may also be stated thus : It is required to ascertain the 
single screw, a twist about which would produce the same effect as any two 
given twists. We shall first examine a special case, and from it we shall 
deduce the general solution. 

Take, as axes of x and y, two screws a, /3, intersecting at right angles, 
whose pitches are p a and pp. Let a body receive twists about these screws 
of amplitudes & cos I and 6 sin I. The translations parallel to the coordinate 
axes are p a 6 cos I and p$& sin I. Hence the axis of the resultant twist makes 
an angle I with the axis of x ; and the two translations may be resolved into 
two components, of which (p a cos 2 1 + pp sin 2 /) is parallel to the axis of the 
resultant twist, while 6 sin I cos I (p a - pp) is perpendicular to the same line. 
The latter component has the effect of transferring the resultant axis of the 
rotations to a distance sin I cos I (p a pp), the axis moving parallel to itself 
in a plane perpendicular to that which contains a and /3. The two original 
twists about a and /3 are therefore compounded into a single twist of 
amplitude & about a screw 6 whose pitch is 



The position of the screw 6 is defined by the equations 

y x tan /, 
z = (p a pp) sin I cos I. 

Eliminating I we have the equation 



22 



20 THE THEORY OF SCREWS. [13, 

The conoidal cubic surface represented by this equation has been called 
the cylindroid*. 

Each generating line of the surface is conceived to be the residence of a 
screw, the pitch of which is determined by the expression 



When a cylindroid is said to contain a screw, it is not only meant that the 
screw is one of the generators of the surface, but that the pitch of the screw 
is identical with the pitch appropriate to the generator with which the screw 
coincides. 

We shall first show that it is impossible for more than one cylindroid to 
contain a given pair of screws 6 and <f>. For suppose that two cylindroids 
A and B could be so drawn. Then twists about 6 and <f> will compound 
into a twist on the cylindroid A and also on the cylindroid B ( 14). There 
fore the several screws on A would have to be identical with the screws on B, 
i.e. the two surfaces could not be different. That one cylindroid can always 
be drawn through a given pair of screws is proved as follows. 

Let the two given screws be 6 and <, the length of their common perpen 
dicular be h, and the angle between the two screws be A ; we shall show that 
by a proper choice of the origin, the axes, and the constants p a and pp, a 
cylindroid can be found which contains 6 and <. 

If I, m be the angles which two screws on a cylindroid make with the 
axis of oc, and if z ly z*, be the values of z, we have the equations of 
which the last four are deduced from the first six 

* This surface has been described by Pliicker (Neue Geometric des Eaumes, 1868-9, p. 97) ; he 
arrives at it as follows : Let ft = 0, and ft = be two linear complexes of the first degree, then all 
the complexes formed by giving /* different values in the expression O + /ufi = form a system of 
which the axes lie on the surface z (x^ + y 2 ) - (k - k )xy = Q. The parameter of any complex of 
which the axis makes an angle w with the axis of x is & = fc cos 2 w + fc sin 2 w. Pliicker also con 
structed a model of this surface. 

Pliicker does not appear to have noticed the mechanical and kinematical properties of the 
cylindroid which make this surface of so much importance in Dynamics ; but it is worthy of 
remark that the distribution of pitch which is presented by physical considerations is exactly 
the same as the distribution of parameter upon the generators of the surface, which Pliicker 
fully discussed. 

The first application of the cylindroid to Dynamics was made by Battaglini, who showed that 
this surface was the locus of the wrench resulting from the composition of forces of varying ratio 
on two given straight lines (Sulla serie dei sistemi di forze, Eendic. Ace. di Napoli, 1869, p. 133). 
See also the Bibliography at the end of this volume. 

The name cylindroid was suggested by Professor Cayley in 1871 in reply to a request which 
I made when, in ignorance of the previous work of both Pliicker and Battaglini, I began to 
study this surface. The word originated in the following construction, which was then 
communicated by Professor Cayley. Cut the cylinder x i + y"=(p -p^ in an ellipse by the 
plane z x, and consider the line x = Q, y=P B ~P a - If any plane z~c cuts the ellipse in the 
points A, B and the line in C, then CA, CB are two generating lines of the surface. 



14] THE CYLINDROID. 21 

Pe = Pa cos 2 / + p ft sin 2 1, z 1 = (p a pp) sin I cos I, 
P<t> = P* C s 2 wi + Pft sin 2 m, z z = (p a pp) sin m cos m, 
A = I m, k = z l z z , 



_ 

Pa JJB A > 

sin A 

Pa + pp=pe+p<t,-h cot A, 

+ 5, 

A 

with similar values for m and z. 2 . It is therefore obvious that the cylindroid 
is determined, and that the solution is unique. 

It will often be convenient to denote by (6, <) the cylindroid drawn 
through the two screws 6 arid <. 

On any cylindroid there are in general two but only two screws which 
like a. and /3 intersect and are at the same time at right angles. These two 
important screws are often termed the principal screws of the surface. 

14. General Property of the Cylindroid. 

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 have 
regained the same position that it held before the first. 

The proof of this theorem must, according to ( 12), 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-corresponding screws, then the three wrenches 
equilibrate. 

The former of these properties of the cylindroid is thus proved : Take 
any three screws 0, </>, i/r, upon the surface which make angles I, m, n, with 
the axis of x, and let the body receive twists about these screws of amplitudes 
, </> , -\Jr . Each of these twists can be decomposed 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. 

The rotations are through angles equal respectively to 

cos I 4- <f> cos m + ty cos n 

and sin I + </> sin m + ty sin n. 

The translations are through distances equal to 

p a (0 f cos I + < cos ra + ->/r cos n) 
and pp (0 sin I + </> sin m + -fy sin n}. 



22 



THE THEORY OF SCREWS. 



[14- 



These four quantities vanish if 

u <p >lr 

sin (w - n) sin (n I) ~ sin (I m) 
and hence the fundamental property of the cylindroid has 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 pro 
vides for the composition 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 ampli 
tudes (or intensities). 

15. Particular Cases. 

If p* =P0 the cylindroid reduces to a plane, and the pitches of all the 
screws are equal. If all the pitches be zero, then the general property of the 
cylindroid reduces to the well-known construction for the resultant of two 
intersecting forces, or of rotations about two intersecting axes. If all the 
pitches be infinite, the general property reduces to the construction for the 
composition of two translations or of two couples. 

16. Cylindroid with one Screw of Infinite pitch. 

Let OP, Fig. 2, be a screw of pitch p about which a body receives a small 
twist of amplitude o>. 




Fig. 2. 



Let OR be the direction in which all points of the rigid body are trans 
lated through equal distances p by a twist about a screw of infinite pitch 



16] 



THE CYLINDROID. 



23 



parallel to OR. It is desired to find the cylindroid determined by these two 
screws. 

In the plane FOR draw OS perpendicular to OP and denote Z ROS 
by X. 

The translation of length p along OR may be resolved into the components 
p sin X along OP and p cos X along OS. 

Erect a normal OT to the plane of POR with a length determined by 
the condition 

coOT = p cos X. 

The joint result of the two motions is therefore a twist of amplitude to 
about a screw 6 through T and parallel to OP. 

The pitch p e of the screw is given by the equation 

(op g = wp + p sin X, 
whence p e p=QT tan X. 




Fig. 3. 



In Fig. 3 we show the plane through OP perpendicular to the plane POR 
in Fig. 2. The ordinate is the pitch of the screw through any point T. 

If p 6 = then OT= OH. Thus H is the point through which the one 
screw of zero pitch on the cylindroid passes, and we have the following 
theorem : 

If one screw on a cylindroid have infinite pitch, then the cylindroid 
reduces to a plane. The screws on the cylindroid become a system of parallel 
lines, and the pitch of each screw is proportional to the perpendicular distance 
from the screw of zero pitch. 



24 THE THEORY OF SCREWS. [17- 

17. Form of the Cylindroid in general. 

The equation of the surface contains only the single parameter p*pp, 
consequently all cylindroids are similar surfaces differing only in absolute 
magnitude. 

The curved portion of the surface is contained between the two parallel 
planes z = (papp), but it is to be observed that the nodal line x = 0, y= 0, 
also lies upon the surface. 

The intersection of the nodal line of the cylindroid with a plane is a 
node or a conjugate point upon the curve in which the plane is cut by the 
cylindroid according as the point does lie or does not lie between the two 
bounding planes. 

18. The Pitch Conic. 

It is very useful to have a clear view of the distribution of pitch upon 
the screws contained on the surface. The equation of the surface involves 
only the difference of the pitches of the two principal screws and one arbitrary 
element must be further specified. If, however, two screws be given, then 
both the surface and the distribution are determined. Any constant added 
to all the pitches of a certain distribution will give another possible distribu 
tion for the same cylindroid. 

Let p g be the pitch of a screw 6 on the cylindroid which makes an angle I 
with the axis of x ; then ( 13) 

Pe = Pa cos 2 1 +pp sin 2 1. 
Draw in the plane x, y, the pitch conic 



where H is any constant ; then if r be the radius vector which makes an angle 
I with the axis of x, we have 

H 
P = -?> 

whence the pitch of each screw on a cylindroid is proportional to the inverse 
square of the parallel diameter of the conic. 

This conic is known as the pitch conic. By its means the pitches of all 
the screws on the cylindroid are determined. The asymptotes, real or 
imaginary, are parallel to the two screws of zero pitch. 

19. Summary. 

We shall often have occasion to make use of the fundamental principles 
demonstrated in this chapter, viz., 



19] 



THE CYLINDROID. 



That one, but only one, cylindroid can always be drawn so that two of its 
generators shall coincide with any two given screws a and ft, and that when all 
the generators of the surface become screws by having pitches assigned to them 
consistent with the law of distribution characteristic of the cylindroid, the pitches 
assigned to the generators which coincide with a and /3 shall be equal to the 
given pitches of a. and /3. 

Thus the cylindroid must become a familiar conception with the student 
of the Theory of Screws. A model of this surface is very helpful, and fortu 
nately there can be hardly any surface which is more easy to construct. In 
the Frontispiece a photograph of such a model is shown, and a plate repre 
senting another model of the same surface will be found in Chap. XIII. 

We shall develop in Chap. V an extremely simple method by which 
the screws on a cylindroid are represented by the points on a circle, and 
every property of the cylindroid which is required in the Theory of Screws 
can be represented by the corresponding property of points on a circle. 



CHAPTER III. 



RECIPROCAL SCREWS. 

20. 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 ft, then conversely a body only free to 
twist about the screw ft will be in equilibrium, though acted upon by a wrench 
on the screw a. 

The principle of virtual velocities states, that if the body be in equili 
brium the work done in a small displacement against the external forces 
must be zero. That the virtual coefficient should vanish is the necessary and 
the sufficient condition, or ( 10) 

(p a + pp) cos d sin = 0. 

The symmetry shows that precisely the same condition is required 
whether the body be free to twist about a, while the wrench act on ft, or 
vice versa. A pair of screws are said to be reciprocal when their virtual co 
efficient is zero. 

21. Particular Instances. 

Parallel or intersecting screws are reciprocal when the sum of their pitches 
is zero. 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 sus 
ceptible of translation. A screw whose pitch is zero or infinite is reciprocal 
to itself*. 

22. Screw Reciprocal to Cylindroid. 

If a screw t] be reciprocal to two given screws 9 and (f>, then 77 is reciprocal 
to every screw on the cylindroid (6, <). 

* See also Professor Everett, F.R.S., Messenger of Mathematics, New Series (1874), No. 39. 



20-23] RECIPROCAL SCREWS. 27 

For a body only free to twist about 77 would be undisturbed by wrenches 
on 6 and < ; but a wrench on any screw ty of the cylindroid can be resolved 
into wrenches on 6 and </> ; therefore a wrench on -^r cannot disturb a body 
only free to twist about 77 ; therefore *fy and rj are reciprocal. We may say 
for brevity that 77 is reciprocal to the cylindroid. 

77 cuts the cylindroid in three points because the surface is of the third 
degree, and one screw of the cylindroid passes through each of these three 
points ; these three screws must, of course, be reciprocal to 77. But 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 I with the axis of x is 

p a cos 3 / +pp sin 2 /. 



This is also the pitch of the screw TT I. There are, therefore, two screws 
of any given pitch ; but there cannot be more than two. It follows that 77 
can at most intersect two screws upon the cylindroid of pitch equal and 
opposite to its own ; and, therefore, 77 must be perpendicular to the third 
screw. Hence any screw reciprocal to a cylindroid must intersect one of the 
generators at right angles. We easily infer, also, that a line intersecting one 
screw of a cyliudroid at right angles must cut the surface again in two 
points, and the screws passing through these points have equal pitch. 

These important results can be otherwise proved as follows. A wrench 
can always be expressed by a force at any point 0, and a couple in a 
plane L through that point but not of course in general normal to the force. 

For wrenches on the several screws of a cylindroid, the forces at any 
point all lie on a plane and the couples all intersect in a ray. 

The first part of this statement is obvious since all the screws on the 
cylindroid are parallel to a plane. 

To prove the second it is only necessary to note that any wrench on the 
cylindroid can be decomposed into forces along the two screws of zero pitch. 
Their moments will be in the planes drawn through arid the two screws of 
zero pitch. The transversal across the two screws of zero pitch drawn from 
must therefore lie in every plane L. 

We hence see that the third screw on the cylindroid which is crossed by 
such a transversal must be perpendicular to that transversal. 

23. Reciprocal Cone. 

From any point P perpendiculars 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 



28 



THE THEORY OF SCREWS. 



[23, 



screws on the cylindroid intersected by the perpendicular, then the perpen 
diculars 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 be the point from which the cone is to be drawn, and through let 
a line OT be drawn which is parallel to the nodal line, and, therefore, perpen 
dicular to all the generators. This line will cut the cylindroid in one real 
point T (Fig. 4), the two other points of intersection coalescing into the in 
finitely distant point in which OT intersects the nodal line. 

Draw a plane through T and through the screw LM which, lying on the 
cylindroid, has the same pitch as the screw through T. This plane can cut 
the cylindroid in a conic section only, for the line LM and the conic will then 




Fig. 4. 

make up the curve of the third degree, in which the plane must intersect the 
surface. Also since the entire cylindroid (or at least its curved portion) is 
included between two parallel planes ( 17), it follows that this conic must be 
an ellipse. 

We shall now prove that this ellipse is the locus of the feet of the per 
pendiculars let fall from 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 
generator of the cylindroid which it meets at V. This generator is, therefore, 
perpendicular to the plane of OT and VT, and, therefore, to the line 0V. 
It follows that V must be the foot of the perpendicular from on the 



24] RECIPROCAL SCREWS. 29 

generator through V, and that, therefore, the cone drawn from 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 cylindroid from any point 0. 

Draw through 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 LM 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 LM will cut the cylindroid in an ellipse, the various points 
of which joined to give the cone required*. 

We may further remark that as the plane TLM passes through a gene 
rator it must be a tangent plane to the cylindroid at one of the intersections, 
suppose L, while at the point M the line LM must intersect another generator. 
It follows (22) that L must be the foot of the perpendicular from T upon LM, 
and that M must be a point upon the nodal line. 

24. 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 generally be confined 
to one ruled surface. But this surface can be no other than a cylindroid. 
For, suppose three screws X, //., v, which were reciprocal to the four given 
screws did not lie on the same cylindroid, then any screw <f> on the cylindroid 
(X, fi), and any screw -^r on the cylindroid (X, v) must also fulfil the conditions, 
and so must also every screw on the cylindroid (<, ^) (22). We should thus 
have the screws reciprocal to four given screws, limited not to one surface, 
as above shown, but to any member of a family of surfaces. The construction 
of the cylindroid which is the locus of all the screws reciprocal to four given 
screws, may be effected in the following manner : 

Let a, /3, 7, 8 be the four screws, of which the pitches are in descending 
order of magnitude. Draw the cylindroids (a, 7) and (13, 8). If <r be a linear 
magnitude intermediate between pp and p y , it will be possible to choose two 
screws of pitch or on (a, 7), and also two screws of pitch <r on (/3, 8). Draw 
the two transversals which intersect the four screws thus selected ; attribute 
to each of these transversals the pitch a, and denote the screws thus pro 
duced by 6, <j>. Since intersecting screws are reciprocal when the sum of their 
pitches is zero, it follows that 6 and </> must be reciprocal to the cylindroids 
(a, 7) and (13, 8). Hence all the screws on the cylindroid (6, </>) must be re 
ciprocal to a, /3, 7, 8, and thus the problem has been solved. 

* M. Appell has proved conversely that the cylindroid is the only conoidal surface for which 
the feet of the perpendiculars from any point on the generators form a plane curve. Revue de 
Mathematiques Speciales, v. 129 30 (1895). More generally we can prove that this property 
cannot belong to any ruled surface whatever except a cylindroid and of course a cylinder. 



30 THE THEORY OF SCREWS. [25-27 

25. Screw Reciprocal to Five Screws. 

The determination of a screw reciprocal to five given screws must in 
general admit of only a finite number of solutions, because the number of 
conditions to be fulfilled is the same as the number of disposable constants. 
It is very important to observe that this number must be unity. For if 
two screws could be found which fulfilled the necessary conditions, then these 
conditions would be equally fulfilled by every screw on the cylindroid 
determined by those screws ( 22), 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 cylindroid, which also 
contains the required screw. These cylindroids must therefore intersect in 
the single screw, which is reciprocal to the five given screws. 

26. Screw upon a Cylindroid Reciprocal to a Given Screw. 

Let e be the given screw, and let X, p, v, p be any four screws reciprocal 
to the cylindroid ; then the single screw 77, which is reciprocal to the five 
screws e, X, //., v, p, must lie on the cylindroid because it is reciprocal to 
X, /A, v, p, and therefore 77 is the screw required. 

The solution must generally be unique, for if a second screw were reciprocal 
to e, then the whole cylindroid would be reciprocal to e ; but this is not the 
case unless e fulfil certain conditions ( 22). 

27. Properties of the Cylindroid*. 

We enunciate here a few properties of the cylindroid for which the writer 
is principally indebted to that accomplished geometer the late Dr Casey. 

The ellipse in which a tangent plane cuts the cylindroid has a circle for 
its projection on a plane perpendicular 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. 

The line joining the points in which the ellipse is cut by two intersecting 
screws on the cylindroid is parallel to the minor axis. 

* For some remarkable quaternion investigations into " the close connexion between the 
theory of linear vector functions and the theory of screws " see Professor C. J. Jolj , Trans. Royal 
Irish Acad., Vol. xxx. Part xvi. (1895), and also Proc. Royal Irish Acad., Third Series, Vol. v. 
No. 1, p. 73(1897). 



CHAPTER IV. 



SCREW CO-ORDINATES. 

28. Introduction. 

We are accustomed, in ordinary statics, 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 ordinary 
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 com 
pounded 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 every way, they shall equilibrate. 

The solution of this problem is identical (12) with that of the problem 
which may be enunciated as follows : 

To determine the amplitudes of seven small twists about seven given screws, 
such that, if these twists be applied to a rigid body in succession, the body 
after the last twist shall have resumed the same position which it occupied 
before the first. 

The problem we 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 a point by each twist may be regarded as rectilinear. Were it 

* If all the pitches be zero, the problem stated above reduces to the determination 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 tetrahedron, we have a problem which has been solved by Mobius, 
Grelle s Journal, t. xvm. p. 207 (1838). 



32 THE THEORY OF SCREWS. [28- 

not for this condition a distinct solution would be required for every variation 
of the order in which the successive twists were imparted. 

If the number of screws were greater than seven, then both problems 
would be indeterminate ; if the number 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, ft, 7, B, e, 77. Find the screw ty which is 
reciprocal to 7, 8, e, , 77. Let the seven wrenches act upon a body only 
free to twist about ty. The reaction of the constraints which limit the 
motion of the body will neutralize every wrench on a screw reciprocal to 
i/r (20). We may, therefore, so far as a body thus circumstanced is con 
cerned, discard all the wrenches except those on a and ft. Draw the 
cylindroid (a, ft), and determine thereon the screw p which is reciprocal to -^r. 
The body will not be in equilibrium unless the wrenches about a and ft 
constitute a wrench on p, and hence the ratio of the intensities a" and ft" 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. (See Appendix, note 1.) 

29. Intensities of the Components. 

Let the six screws of reference be w l , &c. &> 6 , and let p be a given screw 
on which is a wrench of given intensity p". Let the intensities of the 
components be p/ , &c. p 6 ", and let 77 be any screw. A twist about 77 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. If TS^ be 
the virtual coefficient of 7; and the nth screw of reference, we have 

P"^r,p = P"^ + &C. p G "^rfl- 

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 de 
termined. This process will be greatly simplified by judicious choice of 
the six screws of which 77 is the type. Let 77 be reciprocal to G> 2 , &c. &> 6 , then 
oj-,,2 = 0, &c. -or^ = 0, and we have 

P"^r,p ^ Pl ^rft- 

From this equation p" 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 determines the amplitudes of 
the six twists about the six screws of reference into which any given twist 
may be decomposed. 



31 J SCREW CO-ORDINATES. 33 

30. The Intensity of the Resultant may be expressed in terms of the 
intensities of its components on the six screws of reference. 

Let a be any screw of pitch p a , let p lt p. 2 , &c. p e be the pitches of the 
six screws of reference w l} o> 2 , ... &> 6 ; then taking each of the screws of refer 
ence in succession, for 77 in 29, and remembering that the virtual coefficient 
of two coincident screws is simply equal to the pitch, we have the following 
equations : 



ar 6 = ai + + cr-BT,,, + ap s . 
But taking the screw p in place of 77 we have 

<*"pa = l"r ttl + tt/ ^ae. 

Substituting for ^ al ... -5r o6 from the former equations, we deduce 
pj 1 * = $ OW) + 22 (/ V^ 12 ). 

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 of course greatly simplified when the three 
axes are rectangular, and we shall now show how by a special disposition 
of the screws of reference, a corresponding simplification can be made in the 
formula just written. 

31. Co-Reciprocal Screws. 

We have hitherto chosen the six screws of reference quite arbitrarily ; 
we now proceed in a different manner. Take for &) 1? any screw; for co 2 , any 
screw reciprocal to a^; for o> 3 , any screw reciprocal to both &>! and &> 2 ; f r &> 4 , 
any screw reciprocal to o) 1} &>.,, &> 3 ; for &> 5 , any screw reciprocal to a> l , <o 2 > MS, &> 4 ; 
for &> 6 , the screw reciprocal to Wj, &>.,, &> 3 , &> 4 , o> 5 . 

A set constructed in this way possesses the property that each pair 
of screws is reciprocal. Any set of screws not exceeding six, of which each 
pair is reciprocal, may be called for brevity a set of co-reciprocals*. 

Thirty constants determine a set of six screws. If the set be co- 
reciprocal, fifteen conditions must be fulfilled ; we have, therefore, fifteen 
elements still disposable, so that we are always enabled to select a co- 
reciprocal set with special appropriateness to the problem under con 
sideration. 

* Klein Las discussed (Math. Ann. Band n. p. 204 (1869)) six linear complexes, of which each 
pair is in involution. If the axes of these complexes be regarded as screws, of which the 
" Hauptparameter " are the pitches, then these six screws will be co-reciprocal. 

B. 3 



34 THE THEORY OF SCREWS. [31- 

The facilities presented by rectangular axes for questions connected with 
the dynamics of a particle have perhaps their analogues in the conveniences 
which arise from the use of co-reciprocal sets of screws in the present 
theory. 

If the six screws of reference be co-reciprocal, then the formula of the 
last section assumes the very simple form 



32. Co-ordinates of a Wrench. 

We shall henceforth usually suppose that the screws of reference are 
co-reciprocal. We may also speak of the co-ordinates of a wrench*, meaning 
thereby the intensities of its six components on the six screws of reference. 
So also we may speak of the co-ordinates of a twist, meaning thereby the 
amplitudes of its six components about the six scretvs of reference. 

The co-ordinates of a wrench of intensity a" on the screw a are denoted 
by / , ... fi ". The co-ordinates of a twist of amplitude a about a are 
denoted by a/, . . . a B . 

The co-ordinates of a twist-velocity a about a are denoted by a 1? cL, ... d ti . 
The actual motion of the body is in this case a translation with velocity ap a 
parallel to a and a rotation around p with the angular velocity d. 

33. The Work done in a twist of amplitude a about a screw a, by 
a wrench of intensity ft" on the screw ft, can be expressed in terms of the 
co-ordinates. 

Replace the twist and the wrench by their respective components about 
the co-reciprocals. Then the total work done will be equal to the sum of 
the thirty-six quantities of work done in each component twist by each 
component wrench. Since the screws are co-reciprocal, thirty of these 
quantities disappear, and the remainder have for their sumf 



34. Screw Co-ordinates. 

A wrench on the screw a, of which the intensity is one unit, has for its 
components, on six co-reciprocal screws, wrenches of which the intensities 
may be said to constitute the co-ordinates of the screw a. These co-ordinates 
may be denoted by a lt ... a 6 . 

* Pliicker introduced the conception of the six co-ordinates of a system of forces Phil. Trans., 
Vol. CLVI. p. 362 (1866). See also Battaglini, " Sulle dinami in involuzione," Atti di Nnpoli iv., 
(1869); Zeuthen, Math. Ann., Band i. p. 432 (1869). 

t That the work done can be represented by an expression of this type was announced by 
Klein, Math. Ann. Band iv. p. 413 (1871). 



30] SCREW CO-ORDINATES. 35 

When the co-ordinates of a screw are given, the screw itself may be thus 
determined. Let e be any small quantity. Take a body in the position A, 
and impart to it successively twists about each of the screws of reference of 
amplitudes ea ly <z. 2 , ... e 6 . Let the position thus attained be B\ then the 
twist which would bring the body directly from A to B is about the required 
screw a. 

35. Identical Relation. 

The six co-ordinates of a screw are not independent quantities, but 
fulfil one relation, the nature of which is suggested by the relation between 
three direction cosines. 

When two twists are compounded by the cylindroid ( 14), it will be 
observed that the amplitude of the resultant twist, as well as the direction 
of its screw, depend solely on the amplitudes of the given twists, and the 
directions of the given screws, and not at all upon either their pitches or their 
absolute situations. So also when any number of twists are compounded, the 
amplitude and direction of the resultant depend only on the amplitudes and 
directions of the components. We may, therefore, state the following general 
principle. If n twists neutralize (or n wrenches equilibrate) then a closed 
polygon of n sides can be drawn, each of the sides of which is proportional 
to the amplitude of one of the twists (or intensity of one of the wrenches), 
and parallel to the corresponding screw. 

Let a n , b n , c n , be the direction cosines of a line parallel to any screw of 
reference &>, and drawn through a point through which pass three rect 
angular axes. 

Then since a unit wrench on a has components of intensities a,,... a 6 , 
we must have 

(!! 4 . . . + tt (i a fi ) 2 + (&ii + . + &) 2 + (Cjfli 4- . . . + c (i a (i ) 2 = 1, 
whence ^X 2 -f 2Sai<x,cos (12) = 1, 

if we denote by cos (12) the cosine of the angle between two straight lines 
parallel to m 1 and &> 2 . 

36. Calculation of Co-ordinates. 

We may conceive the formation of a table of triple entry from which the 
virtual coefficient of any pair of screws may be ascertained. The three argu 
ments will be the angle between the two screws, the perpendicular distance, 
and the sum of the pitches. These arguments having been ascertained by 
ordinary measurement of lines and angles, the virtual coefficient can be 
extracted from the tables. 

Let a be a screw, of which the co-ordinates are to be determined. The 

32 



36 THE THEORY OF SCREWS. [36- 

work done by the unit wrench on a in a twist of amplitude &>/ about the 
screw &>! is 

2&)j VTal, 

but this must be equal to the work done in the same twist by a wrench of 
intensity j on the screw &> 1} whence 



or j = - . 

Pi 

Thus, to compute each co-ordinate a n , it is only necessary to ascertain 
from the tables the virtual coefficient of e*i and w n and to divide this quantity 



37. The Virtual Coefficient of two screws may be expressed with great 
simplicity by the aid of screw co-ordinates. 

The components of a twist of amplitude are of amplitudes a a lt ... a a B . 

The components of a wrench of intensity ft" are of intensities @" j3 lt ... 
/3"#, 

Comparing these expressions with 32, we see that 



and we find that the expression for the work done in the twist about a, by 
the wrench on ft, is 

a. ft" [2 



The quantity inside the bracket is twice the virtual coefficient, whence we 
deduce the important expression 

l&aft = S/)j !/?]. 

Since a and ft enter symmetrically into this expression, we are again 
reminded of the reciprocal character of the virtual coefficient. 

38. The Pitch of a screw is at once expressed in terms of its co 
ordinates, for the virtual coefficient of two coincident screws being equal 
to the pitch, we have 

p a = 2^ 1 a 1 2 . 

39. Screw Reciprocal to five Screws. 

We can determine the co-ordinates of the single screw p, which is 
reciprocal to five given screws, a, ft, y, 8, e. ( 25.) 

The quantities p l} ... p ti , must satisfy the condition 

, = 0, 



40J SCREW CO-ORDINATES. 37 

and four similar equations ; hence p n p n is proportional to the determinant 
obtained by omitting the /t th column from the matrix or : 



i> 72. 7a. 74. 75> 7 

,, 0,, &,, S 4 , 8 3 , 8 8 

I 
i, e,, e 3 , 6 4 , 6 5 , e 6 , | 

and affixing a proper sign. The ratios of p 1} ... p s , being thus found, the 
actual values are given by 35. 

If there were a sixth screw f the evanescence of the determinant which 
written in the usual notation is (a 1; /3 2 , 73, S 4 , e 5 , f 6 ) would express that the 
six screws had a common reciprocal. This is an important case in view of 
future developments. 

40. Co-ordinates of a Screw on a Cylindroid. 

We may define the screw 8 on the cylindroid by the angle I, which it makes 
with a, one of the two principal screws a and /3. Since a Avrcnch of unit 
intensity on 6 has components of intensities cos I and sin I on a and j3 ( 14), 
and since each of these components may be resolved into six wrenches on 
any six co-reciprocal screws, we must have ( 34) 

6 n = On cos I + fin sin I. 
From this expression we can find the pitch of : for we have 

p e = Spj (: cos I 4- & sin If, 

whence expanding and observing that as a and /3 are reciprocal p 1 a. 1 @ 1 0, 
and also that S,p 1 a l ~ = p a and f!h& >a *Pjh we have the expression already 
given ( 18), viz. 

pe = p* cos 2 I+PP sin 2 1. 

If two screws, and <, upon the cylindroid, are reciprocal, then (m being 
the defining angle of <), 

2p t (! cos I + Pi sin 1) (! cos m + {3 l sin m) = 0, 
or p a cos I cos m+pft sin I sin m = 0. 

Comparing this with 20, we have the following useful theorem : 

Any two reciprocal screws on a cylindroid are parallel to conjugate 
diameters of the pitch conic. 

Since the sum of the squares of two conjugate diameters in an ellipse is 
constant, we obtain the important result that the sum of the reciprocals of the 
pitches of two reciprocal screws on a cylindroid is constant *. 
* Compare Octonions, p. 190, by Alex. M u Aulay, 1898. 



38 THE THEORY OF SCREWS. [41- 

41. The Canonical Co-Reciprocals. 

If all the six screws of a co-reciprocal system are to pass through the same 
point, they must in general constitute a pair of screws of pitches 4- a and 

a on an axis OX, a pair of screws of pitches -1- b and 6 on an axis Y 
which intersects OX at right angles, and a pair of screws of pitches + c and 

c on an axis OZ perpendicular to both OX and OF. 

It is convenient to speak of a co-reciprocal system thus arranged as a set 
of canonical co-reciprocals. The three rectangular axes OX, OY, OZ we may 
refer to as the associated Cartesian axes. 

If a, , 2 2 , ... a 6 be the six co-ordinates of a screw referred to the canonical 
co-reciprocals, then the pitch is given in general by the equation 

p a = a Or - 2 2 ) + b (ot s 2 - a 4 2 ) + c ( 5 2 - 6 2 ). 

It must be remembered that in this formula we assume that the co-ordi 
nates satisfy the condition | 35 

1 = (! + a a ) 2 + (a, + 4 ) 2 + ( 5 + a*) 2 - 

Of course this condition is not necessarily complied with when a.^ ,, ... or 
some of them are infinite, as they are in the case of a screw of infinite 
pitch 44. 

In general the direction cosines of the screw a are 



42. An Expression for the Virtual Coefficient. 

Let X , fjf, v be the direction cosines of the screw B (of pitch p e ) which 
passes through the point x, y , z . Let X", p", v" be the direction cosines of 
the screw a (of pitch p a ) which passes through the point x" , y", z". Then it 
can easily be shown that the virtual coefficient of 6 and a is half the 
expression 

j x - x", y - y\ z - z" 

(l } 9 + P<t>) (X X" + p fJi" + v v") X , fJL , v 

X" , /*" , "" 

43. Equations of a Screw. 

Given the six co-ordinates a l , 2 , . . . 6 of a screw, with reference to a set 
of six canonical co-reciprocals, it is required to find the equations of that screw 
with reference to the associated Cartesian axes. 

If we take for 6 in the expression just written the screw of pitch a in the 
canonical system, thus making 

X = 1 ; pf = ; v = ; x = ; y = ; z = 0, 



44] SCREW CO-ORDINATES. 39 

we have 

2aa l = ( a+p a )\"- (p"z - v"y ), 

similarly 2aa 2 = (- a + p a ) A." (fjif z v"y ), 

we thus find 

\" = a, + a., ; v"y pf z = a (i a,) p a (a, -f a,). 

In like manner we obtain two similar pairs of equations for the required 
equations of the screw a, 

(a.-, + ?) y - (a, f 04) z = a (j - ou) - p a (^ + o 2 ), 



(a, -f a,) z - ( 6 4- ) a; = 6 (a, - a 4 ) - > (a s + 4 ), 



(i). 



(a s + 4 ) # - (i + ,) y=c(a s - a 6 ) - ^ a ( 5 + a 6 

The expressions on the right-hand side of these equations are the co 
ordinates of the extremity of a vector from the origin of length equal to 
the perpendicular distance of a from the centre, and normal to the plane 
containing both a and the origin. 

The co-ordinates of the foot of the perpendicular from the origin on the 
screw a are easily shown to be 

x - ( 5 - 6 ) ( + a 4 ) c - (a 5 + a,) ( 3 - a 4 ) b, 
i/ = (di- a,) ( 3 + O a - (! 4- a ) ( 5 - ) c, 
^ = (a s - a,) (a, -1- a,) b - (a 3 + 4 ) (otj - a 2 ) a, 

44. A Screw of Infinite Pitch. 

The conception of the screw co-ordinates as defined in 41 require special 
consideration in the case of a screw of infinite pitch. Consider a wrench on 
such a screw. If the intensity of the wrench be one unit, then the 
moment of the couple which forms part of the wrench is infinite. As the 
pitches of the screws of reference or any of those pitches are not in general 
to be infinite, it follows that the wrench of unit intensity on a screw of 
infinite pitch must have for its components on one or more of the screws of 
reference wrenches of infinite intensity. 

If therefore a u 2 , ... a 6 be the co-ordinates of a wrench of infinite pitch, 
it is essential that one or more of the quantities ci l} a.,, ... a s shall be 
infinite. 

In the case where the screws of reference form a canonical system we can 
obtain the co-ordinates as follows : 

(p a + a) cos (al) - d Al sin (al) _ (/> a - a) cos (al) - rf al sin (rl) 
~~ ; " 2 ~ - 



40 THE THEORY OF SCREWS. [44- 

It p a be indefinitely great with respect to a and d a i, then 



p a cosf(ai ) pa cos (oci) 

-- 2a "2a 



_ ;j a cos (as) 
" 



26 " " 4 ~ 26 

_ Pa cos (as) _ Pa cos (as) 

a 5 ~ 2c 2c 

If the co-ordinates of a screw not itself at infinity satisfy 

ai + a, = 0; a :j + a 4 = 0; 5 + a 6 = 0; 
then we must have 

for the equations 

(Pa + ) cos (ai) - dai sin (ai) 

ttj = & > 

_ (p a a) cos (ai) - d al sin (ai) 

and two similar pairs could not be otherwise satisfied. 

We are not however entitled to assume the converse, i.e. that if the pitch 
is infinite then the three equations a 1 + a. 2 = 0, &c. must be satisfied. It will 
however be true that 

but some at least of the co-ordinates being infinite, we are in general 
prevented from replacing these equations by the ordinary linear form. 

45. Indeterminate Screw. 

It may however be instructive to investigate otherwise the circumstances 
of a screw a possessing the property that its six co-ordinates a 1( a, ... a^ are 
submitted to the three conditions 

! + a, = ; a s + a 4 = ; a 5 + 6 = 0. 

Two distinct cases must be considered. Either the screw a must have some 
finite points, or it must lie altogether at infinity. The first alternative is now 
supposed. The second will be discussed in the next article. 

If there be any finite points on a then for such points the three left- 
hand members of the equations in 43 are all zero. The three right-hand 
members must also reduce to zero. The only way in which this can be 
accomplished (for we need not consider the case in which all the co-ordinates 
are zero) is by making p a infinite. 



47] SCREW CO-ORDINATES. 41 

The direction of the screw of infinite pitch is indicated by the fact that 
as a twist about it is a translation with components a (i - 2 ), b (a 3 a 4 ), 
c ( g 6 ), the screw must be parallel to a ray of which these three quantities 
are proportional to the direction cosines. 

As the three equations to the screw have disappeared, the situation of 
the screw is indeterminate. This is of course what might be expected, 
because a couple is equally efficacious in any position in its plane. 

46. A Screw at infinity. 

If we have 

<*! + 2 = ; 3 + a 4 = ; 5 + = : 

then the three equations (i), of (43) will be satisfied for a screw entirely at 
infinity, no matter what its pitch may be. From this and the last article we 
see that the three equations 

j + , = ; a 3 + a 4 = ; a, + % = 

may mean either a screw of infinite pitch and indefinite position, or a screw 
of indefinite pitch lying in the plane at infinity. 

47. Screws on one axis. 

The co-ordinates being referred to six canonical co-reciprocals, it is required 
to determine the co-ordinates of the screws of various pitches which lie on the 
same axis as a given screw a. 

We have from 36 

_ (a + j?.)(i + . 2 ) - < sin (al) 

~ 



_ (i + a ) - d ai sin (ai) 

~ 



~~ 
whence ^ w 1 = (^ + tt,). 

2Q 

We thus have the useful results 



6)5 = a& ~ 



These formulae may be verified by observing that one of the equations ( 43) 
defining o> is 

(&> 5 + a> 9 ) y (w 3 + w 4 ) s = a (a^ o> 2 ) p m (coj -f o) a ). 



42 THE THEORY OF SCREWS. [47 

Introducing the values just given for eo this equation becomes 

- (a a + a 4 ) z = a fa - o 2 ) - p a fa + a,), 



as of course it ought to do, for the pitch is immaterial when the question is 
only as to the situation of the screw. 

48. Transformation of Screw-co-ordinates. 

Let i...a 6 be the co-ordinates of a screw which we shall call to, with 
reference to a canonical system of screws of reference with pitches + a and 
a on an axis X ; + b and b on an intersecting perpendicular axis Y, 
and + c and c on the intersecting axis OZ which is perpendicular to both 
OX and Y. 

Let x , 7/0, z be the co-ordinates of any point with reference to the 
associated system of Cartesians. 

Draw through a system of rectangular axes O X , O Y , O Z parallel to 
the original system OX, OF, OZ. 

Let a new system of canonical screws of reference be arranged with pitches 
+ a and a on O X , + b and b on O Y , and + c and c on O Z . 

Let 1} 0., ... # 6 be the co-ordinates of the screw o with regard to these 
new screws of reference. It is required to find these quantities in terms of 

!, ... 6 . 

Let x, y , z be the current co-ordinates of a point on w referred to the 
new axes, the co-ordinates of this point with respect to the old axes being 
x, y, z, 

then x=x + x ; y = y + y - z = z + z< ) . 

The equations of &> with respect to the new axes are ( 43), 

(0, + 0) </ - (&, + OJ z = " (0i - 0.) - JP. (0i + 0,)) 

(0 1 + 0JS-(0 s + 0Jaf = b(0 > -0 4 )-p..(0 t + 0^ ......... (i). 

(0, + 4 ) x - (0, + 0,) y = c(0 5 - 6 ) - p M (0, + 6 }) 

We have also 

( 5 + 8 ) y ~ fa + 4> z = a (tfj - a,) - p u fa + 2 )j 

(! + ,) .2 -fa + a 6 )x = b(a 3 -a,)-p ta (a 3 +a 4 )[ ......... (ii). 

( 3 + ce 4 ) x - fa + a.,) y = c (a g - a fi ) - _p u (a 5 + or 6 )J 

Remembering that the new axes are parallel to the original axes we have 
1 + 3 = a, + a a ; 3 + 4 = a 3 + a 4 ; 5 + 0, = a, + a ti ......... (iii). 



49] SCREW CO-ORDINATES. 43 

Hence by subtracting the several formulae (i) from the formulae (ii) we 
obtain 

// ( 5 + ) - z (flf g + 4 ) = a (a, - a,) - a (0, - 0,)j 

z n (! -I- a,) - a? (a 5 + a fi ) = 6 (a, - a 4 ) - b (0 3 - 4 ) L ........ (iiii). 

# ( 3 + 4> - 2/o (i + 2 ) = c ( s -- a 6 ) - c (0 5 - 6 ) ) 
The six e( {nations (iii) and (iiii) determine 6 l) ... # in terms of ,,.... 

49. Principal Screws on a Cylindroid. 

If two screws are given we determine as follows the pitches of the two 
principal screws on the cylindroid which the two given screws define. 

Let a and /3 be the two given screws. Then the co-ordinates of these 
screws referred to six canonical co-reciprocals are 

!,... and fr,...fr. 

The co-ordinates of any other screw on the same cylindroid are propor 
tional to 

pa, + fr, py 2 + /3,, . . . pa, + & ; 

when p is a variable parameter. 

The pitch p of the screw so indicated is given by the equation ( 41) 
a (pa, + J3J* - a (pa, + /3,) 2 + b (pa 3 + &)* - b (pa, + &) 3 



= p [{p ( ttl + a,) + fr + fr}* + p {(a, + a 4 ) + & + /3 4 j 2 + p {(a s + a) + fr + ft,]- 2 ], 

or 

p*p a + 2pvr ali + p ft = p {p 2 + 2p cos (a/3) +1], 
or 

p- (p a -p) + 2p {CT a/3 - p cos (a/9)] + pp p = 0. 

For the principal screws p is to be determined so that p shall be a maxi 
mum or a minimum ( 18), whence the equation for p is 

Ka0 - p cos (a/3)] 2 = (p a - p) (p ft - p), 
or 

I? sin 2 (a/3) + p (2OT a/3 cos (a/8) - p a - pp) + p a p ft - vr^ = 0. 

The roots of this quadratic are the required values of p. 

The quadratic may also receive the form 

= (P ~ Pa) (p ~ Pft) sin 2 () + d a/5 -sin (a/3) cos (a/9) (/j a 

- i (Pa -p p y cos 2 (a/S) - i #. sin 2 (a/3), 
where d a p is the shortest distance of a and /3. 



44 THE THEORY OF SCREWS, [49, 50 

In this form it i.s obvious that to increase each of the three quantities 
Pa> pp> P by in does not affect the equation. This is of course the well-known 
property of the cylindroid. ( 20.) 

If the quadratic have two equal roots, then all the screws of the cylindroid 
form a plane pencil, and all have equal pitches. 

The discriminant of the quadratic is 

^ a +^) 2 -4w 0(j; a +p0)cos(a/S)-4^ a p0 sin 2 (a/S)+4OT^ 2 si 
- (p a + p ft ) cos (a/3)} 3 + (p a + ppf sin 2 (a/3) - 4>p a p ft sin 2 (a/ 
(^a-^) 2 )sin 2 (a/3). 
Hence discarding the case when (a/3) = we have 

d.fi=0, p*-p ft = 0, 

as was to be expected. 



CHAPTEK V. 

THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE*. 

50. A Plane Representation. 

The essence of the present chapter lies in the geometrical representation 
of a screw by a point. The series of screws which constitute the cylindroid 
correspond to, or are represented by, a series of points in a plane. By 
choosing a particular type of correspondence we can represent the screws 
of the cylindroid by the points of a circle "f. Various problems on the 
cylindroid can then be studied by the aid of the corresponding circle. We 
commence with a very simple process for the discovery of the circle. It will 
in due course appear how this circular representation is suggested by the 
geometry of the cylindroid itself ( 68). 

It has been shown ( 13) that the positions of the several screws on 
the cylindroid may be concisely defined by the intersections of the pairs of 
planes, 

y = x tan 0, 

z = in sin 26. 

In these equations, 9 varies in correspondence with the several screws, while 
in is a parameter expressing the size of the cylindroid. In fact, the whole 
surface, except parts of the nodal line, is contained between two parallel 
planes, the distance between which is 2m. 

The pitch of the screw corresponding to 9 is expressed by 

P=pn + cos 20, 
where p is a constant. 

* See papers in Proceedings of the lioyal Irish Academy, Ser. n. Vol. iv. p. 29 (1883), and the 
Cunningham Memoirs of the lioyal Irish Academy, No. 4 (1886). 

t I may refer to a paper by Professor Mannheim, in the Comptes rendits for 2nd February, 
1885, entitled "Representation plane relative aux deplacements d une figure de forme invariable 
assujettie a quatre conditions." Professor Mannheim here shows how the above plane repre 
sentation might also have been deduced from the instructive geometrical theory which he had 
brought before the Academy of Sciences on several occasions. 



46 THE THEORY OF SCREWS. 

Eliminating 6 between the equations for z and p, we obtain 



[50- 



Let p and z be regarded as the current co-ordinates of a point. Then 
the locus of this point is the circle which forms the foundation of the plane 
representation*. 

7?i is, of course, the radius, and p is the distance of the centre from a 
certain axis. Any point on this circle being given, then its co-ordinates p 
and z are completely determined. Thus sin 20 and cos 20, and, consequently, 
tan 0, are known. We therefore see that the position of a screw and its 
pitch are completely determined when the corresponding point on the circle 
is known. To each point of the circle corresponds one screw on the cylin- 
droid. To each screw on the cylindroid corresponds one point on the circle. 
This may be termed the representative circle of the cylindroid. 

51. The Axis of Pitch. 

Let T (fig. ">) be the origin. Then p Q is the perpendicular ST from the 
centre 8 of the circle to the axis PT. The ordinate AP is the pitch of the 




P T 

Fig. 5. 

* The following elegant construction for the cylindroid is given by Mr T. C. Lewis, Messenger 
of Mathematics, Vol. ix. pp. 15, 1879. " Suppose that a point P moves with uniform velocity 
around a circle while the circle itself rotates uniformly about an axis in its plane with half the 
angular velocity that P has around the centre. Then the perpendiculars from P on the axis of 
rotation trace out the cylindroid, while the lengths of those perpendiculars are the pitches of the 
corresponding screws." This construction is of special interest in connexion with the represen 
tation of the cylindroid by a circle discussed in this chapter. The construction of Mr Lewis shows 
that if the circle rotate around the axis of pitch with half the angular velocity of the point P 
around the circle, then not only does P represent the screw in this circle but the perpendicular 
from P on the axis of pitch is the position of the screw itself. 



52] THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE. 47 

screw, and the line PT may be called the axis of pitch. We have, accord 
ingly, the following theorem : 

The pitch of any screw on the cylindroid is equal to the perpendicular let 
fall on the axis of pitch from the corresponding point on the circle. 

A parallel A A to the axis of pitch cuts the circle in two points, A and 
A , which have equal pitch. The diameter perpendicular to the pitch axis 
intersects the circle in the points U, V of maximum and minimum pitch. 
These points, of course, correspond to the two principal screws on the cylin 
droid. The two screws of zero pitch are defined by the two real or imaginary 
points in which the axis of pitch cuts the circle. 

A fundamental law of the pitch distribution on the several screws of a 
cylindroid is simply illustrated by this geometrical representation. The law 
states that if all the pitches be augmented by a constant addition, the 
pitches so modified will still be a possible distribution. So far as the 
cylindroid is concerned, such a change would only mean a transference of 
the axis of pitch to some other parallel position. The diameter 2m merely 
expresses the size of the cylindroid, and is, of course, independent of the 
constant part in the expression of the pitch. 

52. The Distance between two Screws. 

We shall often find it convenient to refer to a screw as simply equivalent 
to its corresponding point on the circle. Thus, in fig. 6, the two points, A 




and B, may conveniently be called the screws A and B. The propriety of 
this language will be admitted when it is found that everything about a 



48 THE THEORY OF SCREWS. [52- 

screw can be ascertained from the position of its corresponding point on the 
circle. 

Let us, for instance, seek the shortest distance between the two screws 
A and J5. Since all screws intersect the nodal axis of the cylindroid at 
right angles, the required shortest distance is simply the difference between 
the values of m sin 20 for the two screws : this is, of course, the difference 
of their abscissae, i.e. the length PQ. Hence we have the following theorem : 

The shortest distance between two screws, A and B, is equal to the pro 
jection of the chord AB on the axis of pitch. 

We thus see that every screw A on the cylindroid must be intersected 
by another screw A, and the chord A A is, of course, perpendicular to 
the axis of pitch. The ray through S, parallel to the axis of pitch, will give 
two screws, L and M. These are the bounding screws of the cylindroid, and 
in each a pair of intersecting screws have become coincident. The two 
principal screws, U and V, lying on a diameter perpendicular to the axis of 
pitch, must also intersect. 

If all the pitches be reduced by p , then the pitch axis passes through 
the centre of the circle, and the case assumes a simple type. The extremities 
of a chord perpendicular to the axis of pitch define screws of equal and 
opposite pitches, and every pair of such screws must intersect. The screws 
of zero pitch will then be the bounding screws, while the two principal 
screws will have pitches +m and m, respectively. 

53. The Angle between two Screws. 

This important function also admits of simple representation by the 
corresponding circle. Let A, B (fig. 7) denote the two screws; then, if 6 
and 6 be the angles corresponding to A and B, 

AST =26; B8T=28 , 

whence ASB = 2(8- 6 }. 

If IT be any point on the circle, then 

AHB = 0-8 , 
and we deduce the following theorem : 

The angle between two screws is equal to the angle subtended in the circle 
by their chord. 

The extremities of a diameter denote a pair of screws at right angles: 
thus, A , in fig. 7, is the one screw on the cylindroid which is at right angles 
to A. The principal screws, f/"aud V, are also seen to be at right angles. 



54] THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE. 49 

The circular representation of the cylindroid is now complete. We see 
how the pitch of each screw is given, and how the perpendicular distance 




and the angle between every pair of screws can be concisely represented. 
We may therefore study the dynamical and kinematical properties of the 
cylindroid by its representative circle. We commence by proving a funda 
mental principle very analogous to an elementary theorem in Statics. 

54. The Triangle of Twists. 

It has been already shown ( 14) that any three screws on the cylindroid 
possess the following property : 

If a body receive twists about three screws, so that the amplitude of 
each twist is proportional to the sine of the angle between the two non- 
corresponding screws, the body, after the last twist, will be restored to where 
it was before the first. 

With the circular representation of the cylindroid we transform this 
theorem into the following: 

If any three screws, A, B, C (Fig. 8), be taken on the circle, and if twists 
be applied to a body in succession, so that the amplitude of each twist is 
proportional to the opposite side of the triangle ABC, then the body will 
be restored by the last twist to the place it had before the first. 

From the analogy of wrenches, and of twist velocities to twists, we are 
also able to enunciate the following theorems : 

If wrenches upon the three screws A, B, C be applied to any rigid body, 
then these wrenches will equilibrate, provided that the intensity of each is 
proportional to the opposite side of the triangle. 



50 THE THEORY OF SCREWS. [54- 

If twist velocities about the three screws A, B, C animate a rigid body, 
then these twist velocities will neutralize if they are respectively proportional 
to the opposite sides of the triangle. 




55. Decomposition of Twists and Wrenches. 

The theorems we have just enunciated lead to simple rules for effecting 
the composition, or the decomposition of twists or of wrenches. Let a twist 
on a screw X be given, and let it be required to find the components of this 
twist on any two given screws A, B, all three, of course, lying on the same 
cylindroid. Let o be the amplitude of the twist on X. Then, by the last 
article, the following triad of twists on the screws X, A, B, respectively, will 
neutralize : 

BX AX 
w > "AB "AB 

whence the components on A a,nd B of the twist about X are, so far as 
magnitudes are concerned, 

AX 
and "AS 

A similar proposition holds of course for wrenches. 

56. Composition of Twists and Wrenches. 

Let two twists, of amplitudes a and /3, about the screws A and B, respec 
tively, be applied to a rigid body. It is required to find the single resultant 
screw X, and the amplitude CD of the resulting twist. Divide AB (Fig. 9) 
in the point /, so that the segments AI and BI shall be in the inverse ratio 
of a to /3. Bisect the arc AB at H, and draw HI, which will cut the circle 
in the required point X. 

The value of &> is obtained from the equations 



58] THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE. 51 

If the amplitudes a and /3 had opposite signs, then the point / should have 
divided AB externally in the given ratio. 




57. Screw Co-ordinates. 

We have developed in the last chapter the general conception of 
Screw Co-ordinates. In the case of the cylindroid, the co-ordinates of any 
screw X, with respect to two standard scresvs A and B, are found by resolving 
a wrench of unit intensity on X into its two components on A and B. These 
components are said to be the co-ordinates of the screw. If we denote the 
co-ordinates of X by X l and X 2 , we have 

^BX = AX 

l ~AB ~~ AB 

The co ordinates satisfy the identical relation, 



where e denotes the angle between the two screws of reference, that is, the 
angle subtended by the chord AB. 

58. Reciprocal Screws. 

Every screw A on the cylindroid has one other reciprocal screw B tying 
also on the cylindroid ( 26). Denoting as usual A and B by their corre 
sponding points on the circle, we may enunciate the following theorem : 

The chord joining a pair of reciprocal screws passes through the pole of 
the axis of pitch. 

The condition that two screws shall be reciprocal is 
(p a -I- pft) cos 6 d af } sin = 0, 

where p a and p^ are the pitches, 6 is the angle between the two screws, 
and d^ their shortest distance. It is easy to show that this condition 
is fulfilled for any two screws A and B (Fig. 10), whose chord passes through 
0, the pole of the axis of pitch PQ. 

42 



52 



THE THEORY OF SCREWS. 



[58- 



Since SO . ST = >SM 2 = SB 2 , we have Z STA = Z SAB, and z 
whence Z A TB is bisected by ST, and therefore 

Z A TP = \ Z 4 SB = 6 = Z B TQ. 



= z 




It follows that .4Pcos0 = PTs mO, since each is equal to the perpen 
dicular from P on A T. 

Similarly, 

BQ cos = QT sin ; 

whence ( A P + BQ) cos0-(PT + QT) sin = 0, 

which reduces to 

(p a + pi) cos 6 daft sin 6 = 0. 
The theorem has thus been proved. 

We have, therefore, a simple construction for finding the screw B reci 
procal to a given screw A. It is only necessary to join A to 0, the pole 
of the axis of pitch, and the point in which this cuts the circle again gives 
B the required reciprocal screw. 

We also notice that the two principal screws of the cylindroid are reci 
procal, inasmuch as their chord passes through 0. 

59. Another Representation of the Pitch. 

We can obtain another geometrical expression for the pitch, which will be 
often more convenient than the perpendicular distance from the point to the 
axis of pitch. 

Let A (Fig. 11) be the point of which the pitch is required. Join 
AOB, draw AP perpendicular to the axis of pitch PT, and produce AP to 



60] THE REPRESENTATION OF THE CYL1NDROID BY A CIRCLE. 53 

intersect BT at E. Then, since is the pole of PT, the line PT bisects the 
angle ATE ( 58), and therefore AE must be bisected at P. 




From similar triangles, 

OB:AB:: OT : AE; 
whence, if p a be the pitch of A, arid, of course, equal to AP, or \AE, 

_AB.OT 
IP*- QB . 

But since the quadrilateral ASBT is inscribable in a circle, 

OT.OS = OA.OB; 
whence, eliminating OT, we have, finally, 

_ AO.AB 

Pa m<r : 

as OS is constant, we see that p a varies as AO . AB, whence the following 
theorem : 

If AB be any chord passing through 0, the pole of the axis of pitch, then 
the pitch of the screw A is proportional to the product AO . AB. 

60. Pitches of Reciprocal Screws. 

It is known that the sum of the reciprocals of the pitches of a pair of 
reciprocal screws on the cylindroid is constant ( 40). This is also plain 
from the geometrical representation. For, since the triangles APT and BQT 
(Fig. 10) are similar, we have 

AP-.BQ:: TP : TQ :: OA :OB; 



54 THE THEORY OF SCREWS. [60- 

whence is the centre of gravity of particles of masses and placed at 
A and B, respectively. 

From the known property of the centre of gravity, 
.1 1 /I 

Pa Pft \Pa 

but each of the terms on the left-hand side is unity, whence, as required, 

I 1- _?_ 

The second mode of representing the pitch also verifies this theorem. 
For since ( 59) 

AO.AB 



_BO.BA 

Pft ~ ~20S~ ; 
we have 

_AB> AB 2 .AO.BO 

p+pft- 

from which 



but OA . OB is constant for every chord through ; and, as OS is constant, 
it follows that the sum of the reciprocals of the pitches of two reciprocal 
screws on any cylindroid must be constant. 

61. The Virtual Coefficient. 

Let A and B (Fig. 12) be the two screws. Let, as usual, be the pole of 
the axis of pitch PT. Let be the point in which the chord AB intersects 
OT the perpendicular drawn from to the axis of pitch, and let FT be the 
polar of , which is easily shown to be perpendicular to SO. From T let fall 
the perpendicular TF upon AT , and from let fall the perpendicular OG 
upon AB. 



As before ( 58), we have AT F = tT TF=e\ also, since 

^SAO =^AT , and ^SAO = ^ATO, 

we must have ^SAO -^SAO = /.AT O 1 -^ATO, or Z OAG = z TAF- 
whence the triangles OA G and TAF are similar, and, consequently, 



(52] THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE. 55 

but, as in 58, we have 

(p a -TT + pp- TT ) cos - d aft sin = ; 




Fig. 12. 



whence the virtual coefficient is simply, 



AS 

OS 



and we have the following theorem : 

The virtual coefficient of any pair of screws varies as the perpendicular 
distance of their chord from the pole of the axis of pitch. 

We also notice that the line TF expresses the actual value of the virtual 
coefficient. 

The theorem of course includes, as a particular case, that property of 
reciprocal screws, which states that their chord passes through the pole of the 
axis of pitch ( 58). 

62. Another Investigation of the Virtual Coefficient. 

It will be instructive to investigate the theorem of the last article by a 
different part of the theory. We shall commence with a proposition in ele 
mentary geometry. 

Let ABC (Fig. 13) be a triangle circumscribed by a circle, the lengths of 
the sides being, as usual, a, b, c. Draw tangents at A, B, C, and thus form 
the triangle XYZ. It can be readily shown that if masses a 2 , b", c" be placed 



56 



THE THEORY OF SCREWS. 



[62, 

at A, H, C, their centre of gravity must lie on the three lines AX, BY, GZ. 
These lines must therefore be concurrent at /, which is the centre of gravity 
of the three masses. 




Fig. 13. 

Let B Y intersect the circle again at H. Then, since AC is the polar of 
Y, the arc AC is divided harmonically at H and B: consequently the four 
points A, C, B, H subtend a harmonic pencil at any point on the circle. Let 
that point be B, then BC, BI, BA, BZ form a harmonic pencil ; hence CZ is 
cut harmonically, and consequently Z must be the centre of gravity of 
particles, + a? at A, + b 2 at B, and - c 2 at C. 

Suppose the axis of pitch to be drawn (it is not shown in the figure), and 
let h be the perpendicular let fall from Z on this axis, also let p lt p 2 , p 3 be the 
pitches of the screws A, B, C. 

Then, by a familiar property of the centre of gravity, we must have 

Pitf + p^ - p.? = (a? + b 3 - c 2 ) k = 2abh cos C. 

We shall take A, B as the two screws of reference, and if p, and p,, be the co 
ordinates of C with respect to A and B ; then, from the principles of screw 
co-ordinates ( 30), we have 



where W|3 is the virtual coefficient of A and B. In the present case we have 

_ a _b 
pl ~c p *~c 



whence 
and, finally, 



Pi 2 



12 = h cos C. 



63] THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE. 57 

The negative sign has no significance for our present purpose, and hence we 
have the following theorem : 

The virtual coefficient of two screws is equal to the cosine of the angle 
subtended by their chord, multiplied into the perpendicular from the pole of the 
chord on the axis of pitch. 

This is, perhaps, the most concise geometrical expression for the virtual 
coefficient. It vanishes if the perpendicular becomes zero, for then the 
chord must pass through the pole of the pitch axis, and the two screws be 
reciprocal. The cosine enters the expression in order that its evanescence, 
when 0= 90, may provide for the circumstance that the perpendicular is then 
infinite. 

This result is easily shown to be equivalent to that of the last article by 
the well-know ti theorem : 

If any two chords be drawn in a circle, then the cosine of the angle sub 
tended by the first chord, multiplied into the perpendicular distance from its 
pole to the second chord, is equal to the cosine of the angle subtended 
by the second chord, multiplied into the perpendicular from its pole to 
the first chord. 

It follows that the virtual coefficient must be equal to the perpendicular 
from the pole of the axis of pitch upon the chord joining the two screws, 
multiplied into the cosine of the augle in the arc cut off by the axis of pitch. 
This is the expression of 61, namely, 

nr AS 
OG 08 

63. Application of Screw Co-ordinates. 

It will be useful to show how the geometrical form for the virtual coefficient 
is derived from the theory of screw co-ordinates. Let a }> 2 > and ft l} #> be the 
co-ordinates of two screws on the cylindroid ; then, if the screws of reference 
be reciprocal, the virtual coefficient is ( 37) 



Let A, B (Fig. 14) be the screws of reference, and let C and C be the two 
screws of which the virtual coefficient is required. Let PQ be the axis of 
pitch of which is the pole, then lies on AB, as the two screws of reference 
are reciprocal ( 58). 

As AB is divided harmonically at and H, we have 

AO : OB :: HA : HE :: AP : BQ :: p, : p 2 ; 

whence is the centre of gravity of masses , at A and B, respectively. 



58 



THE THEORY OF SCREWS. 



[63- 



If, therefore, AX, BY, OG be perpendiculars on CO , we have, from the 
principle of the centre of gravity, 




Fig. 14. 

or, p 2 AX + PI BY = ( Pl +PJ OG ; 

but, by a well-known property of the circle, if ra be the radius, 

2mAX = AC . AC -, 2mBY = BC.BC - 

whence 



or 



P1 BC.BC + p.,AC.AG = 2m ( Pi + P ,) OG = m ~ ( 60), 

Ob 



BC_ BW_ AC_ 

pl +P - 



OG 



_ _ _ 

AB AB - AB AB~ m OS 



But, from the expressions for screw co-ordinates ( 57), this reduces to 

* 



The required expression has thus been demonstrated. 

We can give another proof of this theorem as follows : 

If the two screws of reference be reciprocal, and if p l and p 2 be the co 
ordinates of another screw, then it is known, from the theory of the co 
ordinates, that the virtual coefficients of this screw, with respect to the screws 
of reference, are p 1 p l and p.,p, respectively ( 37). 

Thus (Fig. 15) the virtual coefficient of X and A must be ( 57), 

BX 



65] THE REPRESENTATION OF THE CVLINDRO1D BY A CIRCLE. 59 

AO.AB 



but we know ( 59) p l 

whence the virtual coefficient is 



280 



AO.BX 2m sin A . AO OG 



= in 



280 2SO OS 

as already determined. This is an instructive proof, besides being much 
shorter than the other methods. 





Fig. 15. Fig. 16. 

64. Properties of the Virtual Coefficient. 

If the virtual coefficient be given then the chord envelopes a circle with 
its centre at the pole of the axis of pitch. 

Two screws can generally be found which have a given virtual coefficient 
with a given screw. 

Let A (Fig. lo) be a given screw, and X a variable screw ; then their 
virtual coefficient is proportional to OG, and therefore to the sine of A, that 
is, to the length BX. Thus, as X varies, its virtual coefficient with A 
varies proportionally to the distance of X from the fixed point B. 

65. Another Construction for the Pitch. 

As the virtual coefficient of two coincident screws is equal to their pitch, 
we shall obtain another geometrical construction for the pitch by supposing 
two screws to coalesce. For (in Fig. 1C), let AG be the chord joining the two 
coincident screws, that is the tangent, then, from 61, we have for the pitch, 

OG 

m OS 
whence the following theorem : 

The pitch of any screw is proportional to the perpendicular on the tangent 
at the point let fall from the pole of the axis of pitch. 



60 THE THEORY OF SCREWS. [66- 

66. Screws of Zero Pitch. 

A screw of zero pitch is reciprocal to itself. The tangent at a point 
corresponding to a screw of zero pitch, being the chord joining two reciprocal 
screws, must pass through the pole of the axis of pitch. This is, of course, 
the same thing as to say that the axis of pitch intersects the circle in two 
screws, each of which has zero pitch. 

67. A Special Case. 

We have supposed that the axis of pitch occupies any arbitrary position. 
Let us now assume that it is a tangent to the representative circle. This 
specialization of the general case could be produced by augmenting the 
pitches of all the screws on the cylindroid, by such a constant as shall make 
one of the two principal screws have zero pitch. 

The following properties of the screws on the cylindroid are then 
obvious : 

1. There is only one screw of zero pitch, 0. 

2. The pitches of all the other screws have the same sign. 

3. The maximum pitch is double the radius. 

4. The screw is reciprocal to every screw on the surface, arid this 
is the only case in which a screw on the cylindroid is reciprocal to every 
other screw thereon. 

68. A Tangential Section of the Cylindroid*. 

Let the plane of section be the plane of the paper in Fig. 17, and let the 
plane contain one of the screws of zero pitch OA. Let OH be the projection 
of the nodal axis on the plane of the paper. Then OA being perpendicular to 
the nodal axis must be perpendicular to OH, Let P be the point where the 
second screw of zero pitch cuts the curve. Then since any ray through P 
and across AO, meets two screws of equal pitch, it must be perpendicular 
to the third screw which it also meets on the cylindroid ( 22). Hence 
PH is perpendicular to the screw through H, and as the latter lies in the 
normal plane through OH it follows that the angle at H is a right angle. 

Any chord perpendicular to AO must for the same reason intersect two 
screws of equal pitch, and therefore APHO must be a rectangle. 

If tangents be drawn at A and P intersecting at T, then it can be shown 
that any chord TLM through T cuts the ellipse in points L and M on two 
reciprocal screws. 

* For proofs of theorems in this article see a paper in the Transactions of the Royal Irish 
Academy, Vol. xxix. pp. 132 (1887). 



68] THE REPRESENTATION OF THE CYLINDROID BY A CIRCLE. 61 

If from any point X a perpendicular XR be let fall on AP, then the pitch 
of the screw through X is XR tan 6, where sin 6 is the eccentricity of the 
ellipse. Also 6 is the angle between the normal to the plane and the nodal 



axis. 




Fig. 17. 

Let two circles be described, one with the major axis of the ellipse as 
diameter and the other with the line joining the two foci as a diameter. Let 
^i be the point in which the ordinate through X meets the first circle and 
X., be the point in which a ray drawn from X^ to the centre meets the second 
circle. Then this point X 2 on this inner circle will be exactly the circular 
representation of the screws on the cylindroid with which this chapter com 
menced. There is only one such circle, for the distance between the foci is 
the same for every tangential section, and so is the distance from the centre 
to the axis of pitch. 



CHAPTER VI. 



THE EQUILIBRIUM OF A RIGID BODY. 



69. A Screw System. 

To specify with precision the nature of the freedom enjoyed by a rigid 
body, it is necessary to ascertain all the screws about which the constraints 
will permit the body to be twisted. When the attempt has been made for 
every screw in space, the results will give us all the information conceivable 
with reference to the freedom of the body, and also with reference to the 
constraints by which the movement may be hampered. 

Suppose that by these trials, n screws A 1} ... A n have been found about 
each of which the body can receive a twist. It is evident, without further 
trial, that twisting about an infinite number of other screws must also be 
possible (w>l): for suppose the body receive any n twists about A^, ... 
A n the position attained could have been reached by a twist about some 
single screw A. The body can therefore twist about A. 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 
twisting must be possible. All these screws, together with A,, ... A n , will in 
general form what we call a screw system of the nth, order. 

If it be found that the body cannot be twisted about any screw which 
does not belong to the screw system of the nth order, then the body is said 
to have freedom of the nth order. It is assumed that A lf ... A n are 
independent screws, i.e. not themselves members of a screw system of order 
lower than n. If this were the case, the screws about which the body could 
be twisted would only consist of the members of that lower screw system. 

Since the amplitudes of the n twists about AI, ... A n are arbitrary, it 
might be thought that there are n disposable quantities in the selection of a 
screw S from a screw system of the ?ith order. It is, however, obvious from 
14 that the determination of the position and pitch of S depends only upon 



69-72] THE EQUILIBRIUM OF A RIGID BODY. 63 

the ratios of the amplitudes of the twists about A 1} ... A n and hence in the 
selection of a screw from the screw system of the nth order, we have n 1 
disposable quantities. 

70. 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 requirements 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 system is appropriate. 

The study of the six types of screw system 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 relegate 
the statics and kinematics to different chapters. We shall not require any 
detailed examination of the constraints. Every conceivable condition of 
constraints must have been included when the six screw systems have been 
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 mentioning the order of the freedom permitted to the body. 

71. Screw Reciprocal to a System. 

If a screw X be reciprocal to n independent screws, A l} ... A n , of a screw 
system of order n, then X is reciprocal to every other screw A which belongs 
to the same screw system. For, by the property of the screw system, it 
appears that twists of appropriate amplitudes about A 1} ... A n , would 
compound into a twist about A. It follows ( 69) that wrenches on A l , ... 
A n , of appropriate intensities ( 30) compound into a wrench on A. Suppose 
these wrenches on A 1} ... A n> were applied to a body only free to twist 
about X, then since X is reciprocal to A lt ... A H , the equilibrium of the 
body would be undisturbed. The resultant wrench on A must therefore be 
incapable of moving the body, therefore A and X must be reciprocal. 

72. The Reciprocal Screw System. 

All the screws which are reciprocal to a screw system P of order n 
constitute a screw system Q of order 6 n. This important 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 n conditions. The screw X has, therefore, 5 n elements still dis 
posable, and consequently (n < f>) an infinite number of screws Q can be 



64 THE THEORY OF SCREWS. [72- 

found which are reciprocal to the screw system P. The theory of reciprocal 
screws will now prove that Q must really be a screw system of order 6 n. 
In the first place it is manifest that Q must be a screw system of some 
order, for if a body be capable of twisting about even six independent screws, 
it must be perfectly free. Here, however, if a body were able to twist about 
the infinite number of screws embodied in Q, it would still not be free, 
because it would remain in equilibrium, though acted upon by a wrench 
about any screw of P. It 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 n, for the number 
of constants disposable in the selection of a screw belonging to a screw 
system is one less than the order of the system ( 36). But we have seen 
that the constants disposable in the selection of X are 5 n, and, therefore, 
Q must be a screw system of order G - n. 

We thus see, that to any screw system P of order n corresponds a reciprocal 
screw system Q of order 6 n. 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 system P. Construct 
6?i screws of the reciprocal system. If then a be reciprocal to these 6 n 
screws, a must in general belong to P. We thus have 6 n conditions to 
be satisfied by any screw when a member of a screw system of order n. 

73. Equilibrium. 

If the screw system 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 system 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 77 
belonging to Q. If the body does not continue at rest, let it commence to 
twist about a. We would thus have a wrench about rj disturbing a body 
which twists about a, but this is impossible, because a and ?; 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 systems, each expresses the locus of a 
wrench which is unable to disturb a body free to twist about any screw of 
the other. 

74. Reaction of Constraints. 

It also follows that the reactions of the constraints by which the move 
ments of a body are confined to twists about the screws of a system P can 
only be wrenches on the reciprocal screw system Q, for the reactions of the 



76] THE EQUILIBRIUM OF A RIGID BODY. 65 

constraints are only manifested by the success with which they resist the 
efforts of certain wrenches to disturb the equilibrium of the body. 

75. Parameters of a Screw System. 

We next consider the number of parameters required to specify a screw 
system of the ?ith order often called for brevity an ?i-system. Since the 
system is defined when n screws are given, and since five data are required 
for each screw, it might be thought that on parameters would be necessary. 
It must be observed, however, that the given 5n data suffice not only for the 
purpose of defining the screw system but also for pointing out n special 
screws upon the screw system, and as the pointing out of each screw on the 
system requires n - 1 quantities ( 69), it follows that the number of 
parameters actually required to define the system is only 

5n n (n 1) = n (6 n). 

This result has a very significant meaning in connexion with the theory 
of reciprocal screw systems 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 system is identical with the number necessary to specify its reciprocal 
screw system. This remark is chiefly of importance in connexion with the 
systems of the fourth and fifth orders, which are respectively the reciprocal 
systems of a cylindroid and a single screw. We are now assured that a 
collection of all the screws which are reciprocal to an arbitrary cylindroid can 
be nothing less than a screw system of the fourth order in its most general 
type, and also, that all the screws in space which are reciprocal to a single 
screw must form the most general type of a screw system of the fifth order. 

76. Applications of Co-ordinates. 

If the co-ordinates of a screw satisfy n linear equations, the screw must 
belong to a screw system of the order 6 n. Let 77 be the screw, and let one 
of the equations be 



whence 77 must be reciprocal to the screw whose co-ordinates are pro 
portional to 

^, . ~ 6 ,(37). 

Pi P* 

It follows that 77 must be reciprocal to n screws, and therefore belong to a 
screw system of order 6 n. 

Let a, /3, 7, B be for example four screws about which a body receives 
twists of amplitudes a , # , 7 , & . It is required to determine the screw p and 
B. 5 



66 THE THEORY OF SCREWS. [76, 

the amplitude p of a twist about p which will produce the same effect as the 
four given twists. We have seen ( 37) that the twist about any screw a 
may be resolved in one way into six twists of amplitudes a a,, ... a 6 , on the 
six screws of reference ; we must therefore have 



p pe = a a,, + /3 @ + 77 fi + 8 8 6 , 
whence p and p l} ... p K can be found ( 35). 

A similar process will determine the co-ordinates of the resultant of any 
number of twists, and it follows from 12 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 
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 system of the ?ith order if it be reciprocal to 
6 n independent screws ; it follows that 6 n conditions must be fulfilled 
when n + 1 screws belong to a screw system of the nib. order. 

To determine these conditions we take the case of n 3, though the 
process is obviously general. Let a, /3, 7, & be the four screws, then since 
twists of amplitudes a, (3 , 7 , & neutralise, we must have p zero and hence 
the six equations 

a a, + /3 A + 77, + S S, = 0, 
&c. 



from any four of these equations the quantities a , /3 , 7 , S can be eliminated, 
and the result will be one of the three 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 system 
of the (n + l)th order, twists of appropriate amplitudes about the screws 
neutralise. The amplitude of the twist about any one screw must be pro 
portional to a function of the co-ordinates of all the other screws. We thus 
see that the evanescence of one function must afford all that is necessary for 
n + 1 screws to belong to a screw system of the nth order.* 

* Philosophical Transactions, 1874, p. 23. 



77] THE EQUILIBRIUM OF A RIGID BODY. 67 

77. Remark on systems of Linear Equations. 

Let a right line be, as usual, represented by the two equations 
Ax + By + Cz + D = 0, 



There are here six independent constants involved, while a right line is 
completely defined by four constants. The fact of course is that these two 
equations not only determine the right line on which our attention is fixed, 
but they also determine two planes through that line. Four constants are 
needed for the straight line and one more for each of the planes, so that there 
are six constants in all. 

If we are concerned with the straight line only the intrusion of two 
superfluous constants is often inconvenient. We can remove them by first 
eliminating y and then x, thus giving the two equations the form 



We have here no more than the four constants P, Q, P , Q , which are indis 
pensable for the specification of the straight line. 

Of course it may be urged that these equations also represent two planes. 
No doubt they do, but the equation z = Px + Q is a plane parallel to the axis 
of y, which is absolutely determined when the straight line is known. The 
plane Aac+By+Cz + D = Q may represent any one of the pencil of planes 
which can be drawn through the straight line. 

Analogous considerations arise when the screws of an n-system are 
represented by a series of linear equations. We commence with the case of 
the two-system, in which of course the screws are limited to the generators 
of a cylindroid. 

Let 0j, 2 , ... 6 be the co-ordinates of a screw referred to any six screws 
of reference. 

Let these co-ordinates satisfy the four linear equations 
A,e, +A&+ +4 6 = 0, 
B& + B,0 2 + ... +B,0 B = 0, 

0,6,+ <7 a 2 + ... + c\e c> = o, 
A0i + A& + ... + A0 8 = o, 

where AI, A, ..., B l , B 2 , ... G ly (7, and D 1} D.,, ... are constants. 

Then it is a fundamental part of the present Theory that the locus so 
defined is a cylindroid (| 76). 

52 



68 THE THEORY OF SCREWS. [77, 78 

But it will be observed that there is here a mass of not fewer than 
20 independent constants, while the cylindroid is itself completely defined by 
eight constants ( 75). The reason is that these four equations really each specify 
one screw, i.e. four screws in all, and as each screw needs five constants 
the presence of 20 constants is accounted for. 

But when it is the cylindroid alone that we desire to specify there is no 
occasion to know these four particular screws. All we want is the system of 
the fourth order which contains those screws. For the specification of the 
position of a screw in a four-system three constants are required. Thus the 
selection of four screws in a given four-system requires 12 constants. These 
subtracted from 20 leave just so many as are required for the cylindroid. 

This is of course the interpretation of the process of solving for 3 ,0 4 , r> , 6 
in terms of 9 and 2 . We get 

3 = P0, + Q0 2 6 4 = P 6, + Q 2 5 = P"0, + Q"0 2 ; 8 6 = P" 0, + Q "0 2 . 

Thus we find that the constants are now reduced to eight, which just serve 
to specify the cylindroid. 

An instructive case is presented in the case of the three-system. The 
three linear equations of the most general type contain 15 constants. But 
a three-system is defined by 9 constants ( 75). This is illustrated by solving 
the equations for 0. 2 , 4 , K in terms of 1; 3 , r ,, when we have 

0, = P0, + Q0 3 + RB S , 
4 = P B, + Q S + R0 5 , 
6 = P"0, + Q"d, + R"0 5 . 

This symmetrical process is specially convenient when the screws of reference 
are six canonical co-reciprocals. 

The general theory may also be set down. An ?i-system of screws is 
defined by 6 n linear equations. These contain 5(6 n) = 30 5n constants. 
We can, however, solve for 6 n of the variables in terms of the remaining n. 
Thus we get 6 n equations, each of which has n constants, i.e. n (6 n) in 
all. This is just the number of constants necessary to specify an ?i-system. 
The original number in the equation 30 - 5n may be written 

n (6 - n) + (6 - n) (5 - n). 

The redundancy of (6 n) (5 n) expresses the number of constants necessary 
for specifying 6 n screws in a system of the (6 n)th order. 



CHAPTER VII. 



THE PRINCIPAL SCREWS OF INERTIA*. 



78. Introduction. 

If a rigid body be free to rotate about a fixed point, then it is well known 
that an impulsive couple about an axis parallel 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 there was on one of the principal 
axes a screw ij with a very small pitch, then a twisting motion about 77 would 
closely resemble a simple rotation about the corresponding axis. An impul 
sive wrench on 77 (i.e. a wrench of great intensity acting for a small time) 
will reduce to a couple when compounded with the necessary reaction of the 
fixed point. If we now suppose the pitch of 77 to be evanescent, we may still 
assert that an impulsive wrench on it] of very great intensity will cause the 
body, if previously quiescent, to commence to twist about ?;. 

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 for a 
body with freedom of the third order of the following general theorem : 

If a quiescent rigid body have freedom of the nth order, then n screws can 
always be found (but not generally more than n), such that if the body receive 
an impulsive wrench on any one of these screws, the body will commence to 
tiuist about the same screw. 

These n screws are of great significance in the present 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, 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. 

* Philosophical Transactions, 1874, p. 27. 



70 THE THEORY OF SCREWS. [79- 

79. Screws of Reference. 

We have now to define the group of six co-reciprocal screws ( 31) which 
are peculiarly adapted to serve as the screws of reference in Kinetic investi 
gations. Let be the centre of inertia of the rigid body, and let OA, OB, 
OC be the three principal axes through 0, while a, b, c are the corresponding 
radii of gyration. Then two screws along OA, viz. w l , &> 2 , with pitches + a, 
a; two screws along OB, viz. o> 3 , &> 4 , with pitches + b, b, and two along 
OC, viz. <y D , w 6) with pitches + c, c, are the co-reciprocal group which 
we shall employ. The group thus indicated form of course a set of canonical 
co-reciprocals ( 41). For convenience in writing the formulae, we shall 
often use p ly ... p 6 , to denote the pitches as before. 

We shall first prove that the six screws thus defined are the principal screws 
of inertia of the rigid body when perfectly free. Let the mass of the body be 
M, and let a great constant wrench on w, act for a short time e. The intensity 
of this wrench is &>/ , and the moment of the couple is aw". We now consider 
the effect of the two portions of the wrench separately. The effect of the force 
&>/ is to give the body a velocity of translation parallel to OA and equal to 

/? 

-rj / . The effect of the couple is to impart an angular velocity d^ about 

the axis OA. This angular velocity is easily determined. The effective 
force which must have acted upon a particle dm at a perpendicular distance 

r from OA is - dm. The sum of the moments of all these forces is 
e 

Mo? -~ . This quantity is equal to the moment of the given couple so that 



60 



Ma 2 - 1 = aw,", 



e 

&> x 



The total effect of the wrench on w l is, therefore, to give the body a 
velocity of translation parallel to OA, and equal to T^WI", and also a velocity 

ff 

of rotation about OA equal to ^ &>/ . These movements unite to form a 

twisting motion about a screw on OA, of which the pitch, found by dividing 
the velocity of translation by the velocity of rotation, is equal to a. This 
same quantity is however the pitch of eoj, and thus it is proved that an 
impulsive wrench on ^ will make the body commence to twist about w^. 
We shall in future represent ew" by the symbol &>/", which is accordingly to 
express the intensity of the impulsive wrench. 



81] THE PRINCIPAL SCREWS OF INERTIA. 7i 

80. Impulsive Screws and Instantaneous Screws. 

If a free quiescent rigid body receive an impulsive wrench on a screw 77, 
the body will immediately commence to twist about an instantaneous screw 
a. The co-ordinates of a being given for the six screws of reference just 
denned, we now seek the coordinates of 77. 

The impulsive wrench on 77 of intensity 77 " is to be decomposed into com 
ponents of intensities tj "r} 1 , ... i} "rj 6 on ( l , ... w s . The component on co n 
will generate a twist velocity about o> H amounting to 

jL W 

M Pn 

but if a be the twist velocity about a which is finally produced, the expression 
just written must be equal to aa n , and hence we have the following useful 
result : 

If the co-ordinates of the instantaneous screw be proportional to a l} ... a fi> 
then the co-ordinates of the corresponding impulsive screw are proportional to 



81. Conjugate Screws of Inertia. 

Let a be the instantaneous screw about which a quiescent body either 
free or constrained in any way will commence to twist in consequence of 
receiving an impulsive wrench on any screw whatever 77. Let ft be the 
instantaneous screw in like manner related to another impulsive screw 

We have to prove that if be reciprocal to a then shall 77 be reciprocal 
to j3. 

When the body receives an impulsive wrench on of intensity "" there 
is generally a simultaneous reaction of the constraints, which takes the form 
of an impulsive wrench of intensity ///" on a screw /*. The effect on the body 
is therefore the same as if the body had been free, but had received an 
impulsive wrench of which the component wrench on the first screw of 
reference had the intensity ^ "^ + p, "^ . This and the similar quantities 
will be proportional to the co-ordinates of the impulsive screw which had the 
body been perfectly free would have /3 as an instantaneous screw. These 
latter, as we have shown in 80, are proportional to p l /3 1 ,p.,@. i ...p 6 /3 6 . Hence 
it follows that, h being some quantity differing from zero, we have 



Multiplying the first of these equations by p^, the second by^oa^, &c. adding 



72 THE THEORY OF SCREWS. [81 

the six products and remembering that a and are reciprocal by hypothesis 
while a. and /* are reciprocal by the nature of the reactions of the constraints, 
we have 



The symmetry of this equation shows that in this case 77 must be reciprocal 
to ft. Hence we have the following theorem which is of fundamental import 
ance in the subject of the present volume. 

If a. be the instantaneous screw about which a quiescent rigid body either 
perfectly free or constrained in any manner whatever commences to twist in 
consequence of an impulsive wrench on some screw tj, and if ft be another 
instantaneous screw, similarly related to an impulsive screw , then whenever % 
is reciprocal to a we shall find that 77 is reciprocal to ft. 

When this relation is fulfilled the screws a and ft are said to be conjugate 
screws of inertia. 

82. The Determination of the Impulsive Screw, corresponding 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 system will, in general, fulfil the required 
condition. 

Let B lf ... B n _ n be 6 n screws selected from the screw system which is 
reciprocal to that corresponding to the freedom of the nth order possessed by 
the rigid body. Let S be the screw about which the body is to twist. Let 
X be any one of the screws, an impulsive wrench about which would make 
the body twist about S ; then any screw Y belonging to the screw system of 
the (7 - n)th order, specified by the screws, X, B 1} ... B 6 _ n is an impulsive 
screw, corresponding to S as an instantaneous screw. For the wrench on Y 
may be resolved into 7 n wrenches on X, B 1} ... B 6 _ n ; of these, all but 
the first are instantly destroyed by the reaction of the constraints, so that the 
wrench on Y is practically equivalent to the wrench on X, which, by hypo 
thesis, will make the body twist about S. 

As an example : if the body had freedom of the fifth order, then an 
impulsive wrench on any screw on a certain cylindroid will make the body 
commence to twist about a given screw. 

As another example : 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. 

83. System of Conjugate Screws of Inertia. 

We shall now show that from the screw system of the nth order P, which 
expresses the freedom of the rigid body, generally n screws can be selected 



84] THE PRINCIPAL SCREWS OF INERTIA. 73 

so that every pair of them are conjugate screws of inertia (81). Let B l , &c. 
B 6 _ n be (6 n) screws defining the reciprocal screw system. Let A l be any 
screw belonging to P. Then in the choice of A l we have n 1 arbitrary 
quantities. Let 7 X be any impulsive screw corresponding to A l as an instan 
taneous screw. Choose A 2 reciprocal to 7 1( B 1} ... B s ^ n , then A l and A 2 are 
conjugate screws, and in the choice of the latter we have n 2 arbitrary 
quantities. Let /., be any impulsive screw corresponding to A 2 as an instan 
taneous screw. Choose A 3 reciprocal to I I , I 2 , B l} ... B 6 _ n , and proceed thus 
until A n has been attained, then each pair of the group A l} &c. A n are 
conjugate screws of inertia. The number of quantities which remain 
arbitrary in the choice of such a group amount to 



or exactly half the total number of arbitrary constants disposable in the 
selection of any n screws from a system of the nth order. 

84. Principal Screws of Inertia. 

We have now to prove the important theorem in Dynamics which affirms 
the existence of n principal Screws of Inertia in a rigid body with n degrees 
of freedom. 

The proof that we shall give is, for the sake of convenience, enunciated 
with respect to the freedom of the third order, but the same method applies 
to each of the other degrees of freedom. 

Let 6 be one of the principal screws of inertia, then an impulsive wrench 
on must make the body commence to twist about 0. In the most general 
case when the body is submitted to constraint, the impulsive wrench on will 
of course be compounded with the reaction on some screw A, of the reciprocal 
system. The result will be to produce the impulsive wrench which would, 
if the body had been free, have generated an instantaneous twist velocity 
about 6. 

We thus have the following equations ( 80) where x and y are unknown : 

pA = uQi + yXi, 



Let a be one of the screws of the three-system in question. Then since X 
must be reciprocal to a we have by multiplying these equations respectively 
by pii, . .. p s ct 6 and adding, 

K = xp^e, + xp.a.,6, + . . . + xp ( &$. 



74 THE THEORY OF SCREWS. 

In like manner if ft, 7 be two other screws of the three-system, 



A = xpijrfi -f 

But as 6 belongs to the three-system its co-ordinates must satisfy three 
linear equations. These we may take to be 



We have thus six linear equations in the co-ordinates of 0. We can therefore 
eliminate those co-ordinates, thus obtaining a determinantal equation which 
gives a cubic for x. 

The three roots of this cubic will give accordingly three screws in the 
three -system which possess the required property. 

Thus we demonstrate that in any three-system there are three principal 
screws of inertia, and a precisely similar proof for each of the six values of 
n establishes by induction the important theorem that there are n principal 
screws of inertia in the screw system of the ?ith order. It is shown in 86 
that all the roots are real. 

We shall now prove that the Principal Screws of Inertia are co-reciprocal. 
Let and </> be two such screws, corresponding to different roots x, x" of 
the equation in x. 

Then we have 



, , , , , 

pi-x p-2 -m p t x 

Let /* be the screw of the reciprocal system on which the impulsive wrench 
is generated by the impulse given on 0. 

Then 

, y/*i , y^ , yn* 

<Pl = // , 0-J = - 7, , 06 = - 7, 

PI X p., x p 6 x 

As fi is reciprocal to and X is reciprocal to <f>, we have 



. , 
/ T n 

Pi - % P* X ps-% 



, , 

// T ~77 "T ~r 

pi - as p. 2 -x p 6 - x 

Subtracting these equations and discarding the factor x x", we get 



__ 
"\ i i / f\ / 



i- x") ( ^ 2 - x )( p 2 - x") ( p 6 - x } ( p s - x"} 



85] THE PRINCIPAL SCREWS OF INERTIA. 75 

which is of course 

2M4>i = ; 

whence 6 and </> are reciprocal, and the same being true for each pair of 
principal screws of inertia we thus learn that they form a co-reciprocal 
system. 

We can also show that each pair of the Principal Screws of Inertia are 
Conjugate Screws of Inertia. 

It is easy to see that 



x ^ Pi^-if^i x v Pi^iA 1 ! \ 

~~i // ** i + 77 j2t - ~ = i , ^ 

& 1 ~- x PI x x x PI -~ x y p^ x \ 

As each of the terms on the left-hand side of this equation is zero, the 
expression on the right-hand is also zero, but this is equivalent to 

whence we show that 6 and </> are conjugate screws of inertia and the 
required theorem has been proved. 

85. An algebraical Lemma. 

Let U and V be two homogeneous functions of the second degree in n 
variables. If either U or V be of such a character that it could be expressed 
by linear transformation as the sum of n squares, then the discriminant 
of U + \V when equated to zero gives an equation of the nth degree in \ 
of which all the roots are real *. 

Suppose that V can by linear transformation assume the form 

x 2 _j_ x 2 -\- x 
and adopt x ly x.,, ... x n as new variables, so that 

TT ,- /vi 2 i ft /-.i 2 i 9/f ** 

U Ct jjwj |~ Lv nJi *2 i~ ^i(rj2 l * / i t * 2 

The discriminant of U+\V will, when equated to zero, give the equation 
for \, 

a n + X, a 12 , . . . a m j = 0, 

; ft 2 i , $2-2 "I" * ^2i 

and the discriminant being an invariant the roots of this equation will be 

* A discussion is found in Zanchevsky, Theory of Screws and its application to Mechanics, 
Odessa, 1889. Mr G. Chawner has most kindly translated the Russian for me. 



76 THE THEORY OF SCREWS. [85, 

the same as those of the original equation. The required theorem will 
therefore be proved if it can be shown that all the roots of this equation 
are real. That this is so is shown in Salmon s Modern Higher Algebra, 
Lesson VI.* 

86. Another investigation of the Principal Screws of Inertia. 

The n Principal Screws of Inertia can also be investigated in the following 
fundamental manner by the help of Lagrange s equations of motion in 
generalized co-ordinates. 

Let . . . be the co-ordinates ( 95) of the impulsive screw. Let 
</>!,... < n be the co-ordinates of the body, then <j>i, ... <f> n will be the co-ordinates 
of the instantaneous screw, and from Lagrange s equations, 



_^ = 
dtdfj d$r 

where T is the kinetic energy and where P-$$i denotes the work done in a 
twist S< t against the wrench. 

If the screws of reference be co-reciprocal and if " be the intensity of 
a wrench on , then 

p, - 2^r&. 

As we are considering the action of only an impulsive wrench the effect of 
which is to generate a finite velocity in an infinitely small time we must 
have the acceleration infinitely great while the wrench is in action. The term 

, - is therefore negligible in comparison with - ( r ) and hence for the 
dfa dt \d<j)J 

impulsive motion -f 

d (dT\ 

*Wr?S 

We may regard i and " as both constant during the indefinitely small time 
e of operation of the impulsive wrench, whence ( 79) 

2f 1 f" = ^. 
Pi cty, 

Hence replacing <fa l} ... (j> n by 6 lt ... B n we deduce the following ( 95, 96). 

If T be the kinetic energy of a body ivith freedom of the nth order, 
twisting about a screw 6 whose co-ordinates referred to any n co-reciprocals 
belonging to the system expressing the freedom are B l} ... n , then the co-ordinates 

* See also Williamson and Tarleton s Dynamics, 2nd edition, p. 457 (1889), and Kouth s 
Rigid Dynamics, Part II, p. 49 (1892). 

t Niven, Messenger of Math., May 1867, quoted by Bouth, Rigid Dynamics, Part I, pp. 327-8. 



86] THE PRINCIPAL SCREWS OF INERTIA. 77 

of an impulsive wrench by which the actual motion of the body could be 
produced are proportional to 

dT ^dT_ 

p l d0 1 " p n dd n 

The existence of n Principal Screws of Inertia can now be readily deduced, 
for suppose that 

l dT l dT 



where X is an unknown factor. If then we make 

T = a u 0!~ + a<v>Q>? + 2,a K 0i0<> . . . 
we have an equation of the nth degree for X as follows : 

tt-ia , . . . a^n =: " 



Ct n i , Gt n2 > &nn 

It is essential to note that T is A function of such a character that by 
linear transformation it can be expressed as the sum of n squares, for suppose 
it could be expressed as 

it would be possible to find a real screw which made H 1} H 2 , ... II n -i each 
zero, and then the kinetic energy of the body twisting about that screw 
would be negative. Of course this is impossible. Hence we deduce from 
85 the important principle that all the Principal Screws of Inertia are real. 

If the equation had a repeated root the number of Principal Screws of 
Inertia is infinite. We take n = 4f, but the argument applies to 3 and 2 
also. (There can be no repeated root when n is either 5 or 6. See chaps. 
XVII. and XVIII.) We can choose variables such that T becomes 

and the pitch X becomes simultaneously 



If therefore the discriminant of T + \p, equated to zero, has a pair of equal 
values for X, we must have a condition like 



Take any screw of the system for which ;i = 0, # 4 = 0, then 

T = M 6* (1*1*0!* + ii*0.?), 
P= p 



, 

or T= p. 

Pl 



78 THE THEORY OF SCREWS. [86- 

Hence we find that for all screws on the cylindroid represented by 6 l} 2 , 0, 
the energy will vary as the pitch when the twist velocity remains the same. 
It appears from the representation of the Dynamical problem in chap. XII. 
that in this case all the screws of the cylindroid O lt 2 , 0, must be principal 
screws of Inertia. The number of principal screws of inertia is therefore 
infinite in this case. (See Routh s Theorem, Appendix, Note 2.) 

87. Enumeration of Constants. 

It is the object of this article to show that there are sufficient constants 
available to permit us to select from the screw system of the ?ith 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 ( 78). 

To prove this, it is sufficient to show that when half the available con 
stants have been disposed of in making the n screws conjugate (81) the 
other half admit of adjustment so as to make the screws also co-reciprocal. 
Choose A! reciprocal to B lt ... B 6 _ n , with n - 1 arbitrary quantities; A z 
reciprocal to A ly B^, ... B K _ n> with n 2 arbitrary quantities, and so on, then 
the total number of arbitrary quantities in the choice of n co-reciprocal 
screws from a system of the ?ith order is 

n(n 1) 
n-l + w-2... +1= -y . 

Hence, by suitable disposition of the n(n 1) constants it might be 
anticipated that we can find at least one group of n screws which are 
both conjugate and co-reciprocal. 

We have now to show that these screws would be the principal screws 
of inertia ( 78). We shall state the argument for the freedom of the third 
order, the argument for any other order being precisely similar. 

Let AI, A 2 , A 3 be the three conjugate and co-reciprocal screws which 
can be selected from a system of the third order. Let B lt B. 2 , B 3 be any 
three screws belonging to the reciprocal screw system. Let R lt R y , R. } be 
any three impulsive screws corresponding respectively to A l} A z , A 2 as 
instantaneous screws. 

An impulsive wrench on any screw belonging to the screw system of the 
4th order defined by R lt B l} B z , B 3 will make the body twist about A, ( 82), 
but the screws of such a system are reciprocal to A 2 and A a ; for since A l and 
A z are conjugate, R l must be reciprocal to A z ( 81), and also to A 3 , since A^ 
and A 3 are conjugate. It follows from this that an impulsive wrench on any 
screw reciprocal to A 2 and A 3 will make the body commence to twist about 
A lt but A l is itself reciprocal to A 3 and A a , and hence an impulsive wrench 



THE PRINCIPAL SCREWS OF INERTIA. 79 

on A 1 will make the body commence to twist about 4,. Hence A l and also 
A. 2 and A 3 are principal screws of inertia. 

We shall now show that with the exception of the ?? screws here deter 
mined, generally no other screw possesses the property. Suppose another 
screw S were to possess this property. Decompose the wrench on S into n 
wrenches of intensities S", ... S n " on A lt ... A n ; this must be possible, 
because if the body is to be capable of twisting about S this screw must 
belong to the system specified by A lt ... A n . The n impulsive wrenches on 
A lt ... 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 $ ... <S n must be proportional to the 
intensities $/ , ... S n ". But if this were the case, then every screw of the 
system would be a principal screw of inertia ; for let X be any impulsive 
screw of the system, and suppose that Y is the corresponding instantaneous 
screw, the components of X on A lt ... A nt have intensities X", ... X n ", 
these will generate twist velocities equal to 

c* C 

Q! y n *-*M -y II 

-~-jj A. i , . . . // -A- n > 
Of 

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 components on the screws 
of reference of the twisting motion about Y must 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 system 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. 

88. Kinetic Energy. 

The twisting motion of a rigid body with freedom of the nth order may 
be completely specified by the twist velocities of the components of the 
twisting motion on any n screws of the system defining the freedom. If the 
screws of reference be a set of conjugate 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 position L, and if the 



80 THE THEORY OF SCREWS. [88- 

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 /*, and that if the body had 
been at rest in the position L, it would have commenced to twist about a 
screw /?, with a kinetic energy E$. 

We are to consider how the amount of energy acquired by the second 
impulse is affected by the circumstance that the body is then not at rest in 
L, but is moving through L in consequence of the former impulse. The 
amount will in general differ from Ep, for the movement of the body may 
cause it to do work against the wrench on JJL 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 
/A will be diminished. Under other circumstances the motion through A 
might be of such a character that the impulsive wrench on p 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 /A. When this is 
the case a and ft must be conjugate screws of inertia ( 81), 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 a , and if the kinetic energy of the same body twisting 
about a screw /3 with a certain twist velocity be Ep, then when the body has 
a motion compounded of the two twisting movements, its kinetic energy will 
amount to E a + Ep provided that a and /3 are conjugate screws of inertia. 

Since this result may be extended to any number of conjugate screws of 
inertia, and since the terms E a , &c., are essentially positive, the required 
theorem has been proved. 

89. Expression for Kinetic Energy. 

If a rigid body have a twisting motion about a screw a, with a twist 
velocity a, what is the expression of its kinetic energy in terms of the 
co-ordinates of a ? 

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 distance. If, therefore, 



90] THE PRINCIPAL SCREWS OF INERTIA. 81 

a body of mass M have a movement of translation with a velocity v its kinetic 
energy expressed in these units is 



The movement is to be decomposed into twisting motions about the 
screws of reference ta l , &c. <o 6 , the twist velocity of the component on w n 
being do,,. One constituent of the twisting motion about <a m consists of 
a velocity of translation equal to dp n a n , and on this account the body 
has a kinetic energy equal to ^Ma 2 p n 2 a n -. On account of the rotation 
around the axis with an angular velocity aa n the body has a kinetic energy 
equal to 

1/J2/V 2 j r2fj m 
2 Lin I / (C///6 

where r denotes the perpendicular from the element dM on co m . Remembering 
that p m is the radius of gyration this expression also reduces to ^Ma?p m *a m 2 , 
and hence the total kinetic energy of the twisting motion about w m is 

Ma?p n *Qi n ~. 

We see, therefore ( 88), that the kinetic energy due to the twisting 
motion about a is 

The quantity inside the bracket is the square of a certain linear mag 
nitude which is determined by the distribution 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 quantity. 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 



90. Twist Velocity acquired by an Impulsive Wrench. 

A body of mass M, which is only free to twist about a screw a, is acted 
upon by an impulsive wrench of intensity ij" on a screw 77. It is required to 
find the twist velocity d which is acquired. 

The initial reaction of the constraints is an impulsive wrench of intensity 
X" on a screw X. Then the body moves as if it were free, but had been acted 
upon by an impulsive wrench of which the component on &> m had the intensity 



This component would generate a velocity of translation parallel to <a n and 
equal to ^ W^n + X^X,,). The twist velocity about &> produced by this 
component is found by dividing the velocity of translation by p n . On the 



B. 



82 THE THEORY OF SCREWS. [90- 

other hand, since the co-ordinates of the screw a are a 1; ... 6 , the twist 
velocity about <u n may also be represented by ay n ( 34), whence 

Iiii 
r\ 



If we multiply this equation by p n 2 w , add the six equations found by giving 
n all values from 1 to 6, and remember that a and X are reciprocal, we 
have ( 39) 

ftua z = M -n"^a\ 

whence a is determined. 

This expression shows that the twist velocity produced by an impulsive 
wrench on a given rigid body constrained to twist about a given screw, varies 
directly as the virtual coefficient and the intensity of the impulsive wrench, 
and inversely as the square of u a . (See Appendix, Note 3.) 

91. The Kinetic Energy acquired by an Impulsive Wrench can 

be easily found by 89 ; for, from the last equation, 

1///0 
fj ^ 

Mr/I ll 2 - __ - _ 7TT - 

(Ua ~ M ?C "" 

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 . 

92. Formula for a free body. 

We shall now express the kinetic energy communicated by the impulsive 
wrench on TJ to the body when perfectly free. The component on eo n of 
intensity r!"t] n imparts a kinetic energy equal to 



///o 



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 15 square terms, of 
which (>!! 770 j9 2 02?7i) 2 is a specimen. The entire expression being therefore 
essentially positive shows that a given impulsive wrench imparts greater 
energy to a quiescent body when free than to the same quiescent body when 
constrained to twist about a certain screw. 



95j THE PRINCIPAL SCREWS OF INERTIA. 83 

93. Lemma. 

If a group of instantaneous screws belong to a system of the rath order, 
then the body being quite free the corresponding group of impulsive screws 
also belong to a system of the nth order ; for, suppose that n + 1 twisting 
motions about n + 1 screws neutralise, then the corresponding n + 1 im 
pulsive wrenches must equilibrate, but this would not be possible unless all the 
impulsive screws belonged to a screw system of the ?ith order. 

94. Euler s Theorem. 

If a free or constrained rigid body is acted upon by an impulsive wrench, the 
body will commence to move with a larger kinetic energy when it is permitted 
to select its own instantaneous screw from the screw system P defining the 
freedom, than it would have acquired, had it been arbitrarily restricted to 
any other screw of the system. 

Let Q be the reciprocal system of the (6 w)th order, and let P be the 
screw system of the nth order, consisting of those impulsive screws which, 
if the body were free, would correspond to the screws of P as instantaneous 
screws. 

Let i) be any screw on which the body receives an impulsive wrench. De 
compose this wrench into components on a system of six screws consisting of any 
n screws from P , and any 6 n screws from Q. The latter are neutralised by 
the reactions of the constraints, and may be omitted, while the former com 
pound into one wrench on a screw belonging to P ; we may therefore 
replace the given wrench by a wrench on , 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 impulse about ", consider the body as being perfectly free. 
It follows from 92 that there would be a loss of energy if the body were 
compelled to twist about any screw other than a, which is the one it naturally 
chooses. 

95. Co-ordinates in a Screw System. 

The co-ordinates of a screw belonging to a specified screw-system can be 
greatly simplified by taking n co-reciprocal screws belonging to the given 
screw system as a portion of the six screws of reference. The remaining 
6 n screws of reference must then belong to the reciprocal screw system. 
It follows that out of the six co-ordinates a l} ... a tt of a screw a, which belongs 
to the system, 6 n are actually zero. Thus we are enabled to give the more 
general definition of screw co-ordinates which is now enunciated. 

If a wrench, of which the intensity is one unit on a screw a, which belongs to 

62 



84 THE THEORY OF SCREWS. [95- 

a certain screw system of the nth order, be decomposed into n wrenches of 
intensities a 1} ... a n on n co-reciprocal screws belonging to the same screw system, 
then the n quantities a 1} . . . ^ are said to be the co-ordinates of the screw a. Thus 
the pitch of a will be represented by p^ + . . . + p n a n *. The virtual coefficient 
of a and /3 will be (p&fa + ... +p n <*n@n)- 

We may here remark that in general one screw can be found upon a screw 
system of the wth order reciprocal to n 1 given screws of the same system. 
For, take 6 n screws of the reciprocal screw system, then the required screw 
is reciprocal to 6 n + n 1 = 5 known screws, and is therefore determined 
( 25). 

96. The Reduced Wrench. 

A wrench which acts upon a constrained rigid body may in general be 
replaced by a wrench on a screiu belonging to the screw system, which defines 
the freedom of the body. 

Take n screws from the screw system of the ?ith order which defines the 
freedom, and 6 n screws from the reciprocal system. Decompose the given 
wrench into components on these six screws. The component wrenches on 
the reciprocal system are neutralized by the reactions of the constraints, and 
may be discarded, while the remainder must compound into a wrench on the 
given screw system. 

Whenever a given external wrench is replaced by an equivalent wrench 
upon a screw of the system 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 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 in general be resolved into two wrenches one on a screw of any 
given system, and the other on a screw of the reciprocal screw system. The 
former of these is what we denote by the reduced wrench. 

This theorem of the reduced wrench ceases to be true in the case when 
the screw system and the reciprocal screw system have one screw in common. 
As such a screw must be reciprocal to both systems it follows that all the 
screws of both systems must be comprised in a single five-system. This is 
obviously a very special case, but whenever the condition indicated is satisfied 
it will not be possible to resolve an impulsive wrench into components on the 
two reciprocal systems, unless it should also happen that the impulsive 
wrench itself belongs to the five-system*. 

* I am indebted to Mr Alex. M c Aulay for having pointed out in his book on Octonions, p. 251, 
that I had overlooked this exception when enunciating the Theorem of the reduced wrench in the 
Theory of Screws (1876). 



97] THE PRINCIPAL SCREWS OF INERTIA. 85 

97. Co-ordinates of Impulsive and Instantaneous Screws. 

Taking as screws of reference the n principal screws of inertia ( 84), we re 
quire to ascertain the relation between the co-ordinates of a reduced impulsive 
wrench and the co-ordinates of the corresponding instantaneous screw. If the 
co-ordinates of the reduced impulsive wrench are v\" , ...r} n ", and those of the 
twist velocity are a l} d 2> ... a n , then, remembering the property of a principal 
screw of inertia ( 78), and denoting by u l} ... u n , the values of the magnitude 
u ( 89) for the principal screws of inertia, we have, from 90, 

^ = M^" 

whence observing that d l = dot 1 ; ... a n = da n we deduce the following theorem, 
which is the generalization of 80. 

If a quiescent rigid body, which has freedom of the nth order, commence 
to twist about a screw a, of which the co-ordinates, with respect to the 
principal screws of inertia, are a lt ... a n and if p 1} ... p n be the pitches, and 
iii, ... u n the constants defined, in 89, of the principal screws of inertia, 
then the co-ordinates of the reduced impulsive wrench are proportional to 

U,* U n * 

!, ... a n . 

PI Pn 

Let T 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 
denoted by a,, a 2 , ... d n . [These screws will not be co-reciprocal unless in the 
special case where they are the principal screws of inertia.] It follows ( 88) 
that the kinetic energy will be the sum of the n several kinetic energies due 
to each component twisting motion. Hence we have ( 89) 



T = Mufa? + . . . + 
and also u a * = u? a, 3 + . . . + u n 2 a 2 . 

Let i, ... a n and /3 1( ... /3 7i , be the co-ordinates of any two screws belong 
ing to a screw system of the ?ith order, referred to any n conjugate screws of 
inertia, whether co-reciprocal or not, belonging to the same screw system, then 
the condition that a and /3 should be conjugate screws of inertia is 

Mi a a x ft + . . . + ujttnpn = 0. 

To prove this, take the case of n = 4, and let A, B, C, D be the four screws of 
reference, and let A l} ... A s be the co-ordinates of A with respect to the six 
principal screws of inertia of the body when free ( 79). The unit wrench on 
a is to be resolved into four wrenches of intensities cti, ... a 4 on A, B, C, D: 
each of these components is again to be resolved into six wrenches on the 



86 THE THEORY OF SCREWS. [97, 98 

screws of reference. The six co-ordinates of a, with respect to the same 
screws, are therefore 

A^j, + B 1 a 2 + Ojttg + D^t, 



A^ + B 6 a, + C 

We can now express the condition that a and /3 are conjugate screws of inertia. 
This condition is ( 81) 

Sp! 2 (A lttl + B^, + 0,0, + A4> (A& + B& + C& + A/3 4 ) - 0. 

Denoting pfAf + ... +p 6 A/ by uf y and observing that S l p 1 2 A l B 1 and similar 
expressions are zero, we deduce 



A similar proof may be written down for each of the remaining degrees 
of freedom. 



CHAPTER VIII. 



THE POTENTIAL. 



98. The Potential. 



Suppose a rigid body which possesses freedom of the ?tth order be sub 
mitted to a system of forces. Let the symbol define a position of the 
body from which the forces would be unable to disturb it. By a twist of 
amplitude & about a screw belonging to the screw system, the body may be 
displaced from to an adjacent position P, the energy consumed in making 
the twist being denoted by the Potential V, and no kinetic energy being 
supposed to be acquired. The same energy would be required, whatever be 
the route by which the movement is made from to P. So far as we are 
at present concerned V varies only with the changes of the position of P 
with respect to 0. The most natural co-ordinates by which the position 
P can be specified with respect to are the co-ordinates of the twist ( 32) 
by which the movement from 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 system defining the freedom of the body, then ( 95) 
there will be only n co-ordinates required, and these may be denoted by 
0i, ... &n. 

The Potential V must therefore depend only upon certain quantities 
independent of the position and upon the n co-ordinates #/, . . . n ; and since 
these are small, it will be assumed that V must be capable of development 
in a series of ascending powers and products of the co-ordinates, whence we 
may write 



+ terms of the second and higher orders, 

where H, H 1} ... H n are constants, in so far as different displacements are 
concerned. 

In the first place, it is manifest that H = ; because if no displacement 
be made, no energy is consumed. In the second place, H l , ... H n must also 
be each zero, because the position is one of equilibrium ; and therefore, 



88 THE THEORY OF SCREWS. [98- 

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 generally a homogeneous function of the second degree of the 
n co-ordinates, by which the displacement is defined. 

99. The Wrench evoked by Displacement. 

When the body has been displaced to P, the forces no longer equilibrate. 
They have now a certain resultant wrench. We 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 ( 96) upon a screw of the 
system, 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 0, 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 system, 
while the intensities of the components on the screws of reference are 
*7i"j W- Suppose that the body be displaced from P to an excessively 
close position P , the co-ordinates of P , with respect to 0, being ( 95) 

0i + 861, . . . n + 80 n . 
The potential V of the position P is 



it being understood that 861, ... 80 n are infinitely small magnitudes of a 
higher order than #/, . . . n . 

The work done in forcing the body to move from P to P is V V. 
This must be equal to the work done in the twists about the screws of 
reference whose amplitudes are S0/, . . . 80 n , by the wrenches on the screws 
of reference whose intensities are T?/ , ... rj n ". As the screws of reference 
are co-reciprocal, this work will be equal to ( 33) 

+ 2r h "p 1 80 1 +... + 2r Jn "pn80 n - 

Since the expression just written must be equal to V V for every 
position P in the immediate vicinity of P, we must have the coefficients of 
80i, ... 80 n equal in the two expressions, whence we have n equations, of 
which the first is 

"_ JL d l 

7/1 : + 2 PI d0\ 
Hence, we deduce the following useful theorem : 



100] THE POTENTIAL. 89 

If a free or constrained rigid body be displaced from a position of equi 
librium by twists of small amplitudes, #, , ... # , about n co-reciprocal screws 
of reference, and if V denote the work done in producing this movement, 
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 with 
respect to the corresponding amplitude, and changing 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 0, are proportional to 

^dT 1 dT 
2 Pl de, "2p n d0 n vs 

100. Conjugate Screws of the Potential. 

Suppose that a twist about a screw 6 evokes a wrench on a screw ?/, 
while a twist about a screw <f> evokes a wrench on a screw . If 6 be reci 
procal to , then must <f> be reciprocal to ij. This will now be proved. 

The condition that 6 and are reciprocal is 

pAti + + p n 6 n n = ; 

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 



but we have seen ( 99) that 

_ 1^ t _ 

*%V<*fc" nn 

whence the condition that 9 and f are reciprocal is 

0/jEj^p9W&** 

d(f)i d<f) n 

Now, as V$ is an homogeneous function of the second order of the quantities 
<J>i, ... (j> n } we may write 

V* = A u ft* + . . . + A nn $n> + 2^ 12 <k> s + 2^/0; + . . . , 

in which A hk = A ai . 
Hence we obtain: 



90 THE THEORY OF SCREWS. [100- 

Introducing these expressions we find, for the condition that and should 
be reciprocal, 

This may be written in the form : 

.. /) i i i . A a > i i i f Q I i f) i >\ I A 

"-ll* ! Y I i " L nn"n 0n T "1J \"l 0-2 T t Qi ) i = " 

But this equation is symmetrical with respect to and <, and therefore 
we should have been led to the same result by expressing the condition that 
was reciprocal to 77. 

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 system of the fifth order, which embraces all the screws reciprocal to 
77, as in a certain sense the locus of <. All the screws conjugate to 0, and 
which, at the same time, belong to the screw system C by which the freedom 
of the body is defined, must constitute in themselves a screw system of the 
(n l)th order. For, besides fulfilling the 6 n conditions which define the 
screw system C, they must also fulfil the condition of being reciprocal to 77 ; 
but all the screws reciprocal to 7 n screws constitute a screw system of the 
(ra-l)th order ( 72). 

The reader will be careful to observe the distinction between two conju 
gate screws of inertia ( 81 ), and two conj ugate 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 distribution of its material, while the latter involve extrinsic 
considerations, arising from the forces to which the body is submitted. 

101. Principal Screws of the Potential. 

We now prove that in general 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 on 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, 99: 

dV a 



i = + , , 
2wj dflj 



and ( /i 1) similar equations. 



102] THE POTENTIAL. 91 

Introducing the value of V a , and remembering (34) that a/ = <*"! and 
a/ = a ci, we have the following n equations: 

(ft \ 
AH rpi) + a 2 A 12 + ... + a n A ln =0, 
a / 

&c., &c. 
aiA m + a. 2 A n2 + ... +a n (A nn T p n j = 0. 

\ ^* / 

From these linear equations a n ... a n can be eliminated, and we obtain 

// " 

an equation of the nth degree in . The values of substituted suc 
cessively in the linear equations just written will determine the co-ordinates 
of the n principal screws of the potential. If the position of equilibrium 
be one which is stable for all displacements then V a must under all 
circumstances be positive. As it can be reduced to the sum of n squares 
all the roots of this equation will be real ( 86) and consequently all the n 
principal screws of the potential will be real. 

We can now show that these n screws are co-reciprocal. It is evident, 
in the first place, that if S be a principal screw of the potential, and if be 
a displacement screw which evokes a wrench on 77, the principle of 100 
asserts that, when 6 is reciprocal to S, then must also y be reciprocal to S. 
Let the n principal screws of the potential be denoted by Si, ... S n , and let 
T n be that screw of the screw system which is reciprocal to Si, ... S n -i ( 95), 
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 system 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 
potential. But there are only n principal screws of the potential, therefore 
T n must coincide with S n , and therefore S n must be reciprocal to Si, ... S n ^. 

102. Co-ordinates of the Wrench evoked by a Twist. 

The work done in giving the body a twist of small amplitude a! about a 
screw a, may be denoted by 

fttfta*. 

In fact, remembering that a a 1 = 1 , ... , and substituting these values for 
0.1 in I 7 " ( 100), we deduce the expression: 

Fv a - = A n a,- + ...+ A nn a n - + 24 18 * 1 a + 24 13 a, + . . . 

where F is independent of a and has for its dimensions a mass divided by the 
square of a time, and where v a is a linear magnitude specially appropriate 
to each screw a, and depending upon the co-ordinates of a, and the constants 
in the function 



92 THE THEORY OF SCREWS. [102, 

The parameter v a may be contrasted with the parameter u a considered 
in 89. Each is a linear magnitude, but the latter depends only upon the 
co-ordinates of a, and the distribution of the material of the rigid body. 
Both quantities may be contrasted with the pitch p a , which is also a linear 
magnitude, but depends on the screw, and neither on the rigid body nor the 
forces. 

If a body receive a twist of small amplitude a? about one of the principal 
screws of the potential, then the intensity of the wrench evoked on the 
same screw is (99) : 

1 dV a 
+ 2p a da 

but we have just seen that V=Fv a -a 2 , whence we have the following 
theorem : 

If a body which has freedom of the nth 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 arc i, ... a n , then a reduced 
wrench ( 96) is evoked on a screw with co-ordinates proportional to 

/tj 2 rt 2 

1 -!,... a n , where v l} ... , p l} ... , are the values of the quantity v, and 

PI Pn 

the pitch p, for the principal screws of the potential. 

We can now express with great simplicity the condition that two screws 
and (f> shall be conjugate screws of the potential. For, if 6 be reciprocal 
to the screw whose co-ordinates are proportional to 



we have : 

Vi*0i<l> l +...+v n *0 n <f> n = 0. 

The expression for the potential assumes the simple form 

Fv 1 *a 1 3 +... + Fv n *a n s . 

If the function V be 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. 



103. 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 n (n 1) conditions, being the total number available in 
the selection of n screws in an n system. 



103] 



THE POTENTIAL. 



93 



We now show that the expression of the potential will consist of the 
sum of n square terms, when referred to any set of n conjugate screws of the 
potential. 

The energy consumed in giving a body a twist of amplitude 6 from the 
position of equilibrium to a new position P, is equal to Fv e *0 z ( 102), 
and 77 is the screw on which the wrench is evoked. Suppose that from 
the position P the body receive a twist of amplitude <j> about a screw <, it 
would generally riot be correct to assert that the energy consumed in the 
second twist was proportional 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 rj, or the energy expended in the second twist 
will be rendered less, in consequence of the assistance afforded by the wrench 
on t]. If, however, 77 be reciprocal to (f>, then the quantity of energy con 
sumed in the twist about </> will be unaffected by the presence of a wrench 
on 77. Hence if 6 and <f> be two conjugate screws of the potential, the 
energy expended in giving the body first a twist of amplitude 6 about 6, 
and then a twist of amplitude </> about <f>, is to be represented by 



By taking a third screw, conjugate to both 6 and <, and then a fourth 
screw conjugate to the remaining three, and so on, we see finally that the 
potential reduces to the sum of n square terms, where each pair of the 
screws of reference are conjugate screws of the potential. 



CHAPTER IX. 



HARMONIC SCREWS. 



104. Definition of an Harmonic Screw. 

We have seen in 97 that to each screw 6 of a screw system of the nth 
order corresponds a screw X, belonging to the same screw system. The 
relation between 6 and X is determined when the rigid body, and also the 
screw system which defines its freedom, are completely known. The physical 
connection between the two screws 6 and \ may be thus stated. If an 
impulsive wrench act on the screw \, the body, if previously quiescent, will 
commence to move by twisting about 6. 

We have also seen ( 102) that to each screw 6 of a screw system of the 
nth order corresponds a certain screw 77 belonging to the same screw system. 
The relation between 6 and 77 is determined when the rigid body, the forces, 
and the screw system which defines the freedom, is known. The physical 
connexion between the two screws 6 and 77 may be thus stated. If the body 
be displaced from a position of equilibrium by a twist about 6, the evoked 
wrench, when reduced ( 96) to the screw system, acts on 17. 

The rigid body being given in a position of equilibrium, and the forces 
which act on the body being known, and also the screw system by which the 
freedom of the body is prescribed, we then have corresponding to each screw 
of the given screw system, two other screws X and 77, which also belong to 
the same screw system. 

Considering the very different physical character of the two systems of 
correspondence, it will of course usually happen that the two screws X and 77 
are not identical. But a little reflection will enable us to foresee what we 
shall afterwards prove, viz., that when 6 has been appropriately chosen, then 
\ and 77 may coincide. For since n 1 arbitrary quantities are disposable in 
the selection of a screw from a screw system of the nth order ( 69), it follows 
that for any two screws (for example X and 77) to coincide, n1 conditions 



104] HARMONIC SCREWS. 95 

must be fulfilled ; but this is precisely the number of arbitrary elements 
available in the selection of 6. We can thus conceive that for one or more 
particular screws 6, the two corresponding screws \ and 17 are identical ; and 
we shall now prove the following important theorem : 

If a rigid body be displaced from a position of equilibrium by a twist 
about a screw 6, and if the evoked wrench tend to make the body commence to 
twist about the same screw 6, then if we call 9 an harmonic screw ( 106), we 
assert that the number of harmonic screws is generally the same as the order 
of the screw system which defines the freedom of the rigid body. 

We shall adopt as the screws of reference the n principal screws of inertia. 
The impulsive screw, which corresponds to 6 as an instantaneous screw, will 
have for co-ordinates 



, 

Pi Pn 

where A is a certain constant which is determined by making the co-ordinates 
satisfy the condition ( 35). If 6 be an harmonic screw, then, remembering 
that the screws of reference are co-reciprocal ( 87), we must have n equations, 
of which the first is ( 102) : 

1 L U *0>- J ^1 

%/ *2 Pl de>- 

j nm 

Assuming -p = Ms", where M is the mass of the body, and .9 an unknown 
quantity, and developing F, we deduce the n equations : 
6, (A n + Ms*u*} + z A K +...+ n A ln = 0, 



t A m + 0*A m +... + n (A nn + Ms 2 u n 2 ) = 0. 

Eliminating 0^, ... n , we have an equation of the nth degree for s". The 
n roots of this equation are all real ( 85), and each one substituted in the 
set of n equations will determine, by a system of n linear equations, the 
ratios of the n co-ordinates of one of the harmonic screws. 

It is a remarkable property of the n harmonic screws that each pair of 
them are conjugate screws of inertia, and also conjugate screws of the 
potential. Let H l , . . . H n _ t , be n 1 of the harmonic screws, to which 
correspond the impulsive screws S lt ... S n ^. Also suppose T to be that one 
screw of the given screw system which is reciprocal to S lt ...$_! ( 95), 
then T must form with each one of the screws H lt ... 7/ n _! a pair of con 
jugate screws of inertia (81). But, since S lt ... /S M _, are the screws on 
which wrenches are evoked by twists about //,,... #_! respectively, it is 



96 THE THEORY OF SCREWS. [104, 

evident that T must form with each one of these screws H lt ... H n ^ a pair 
of conjugate screws of the potential ( 100). It follows that the impulsive 
screw, corresponding to T as the instantaneous screw, must be reciprocal to 
7/i , ... ff n -i , and also that a twist about T must evoke a wrench on a screw 
reciprocal to H lt ... H n ^. As in general only one screw of the system can 
be reciprocal to H l ,...H n ^, it follows that the impulsive screw, which 
corresponds to T as an instantaneous screw, must also be the screw on which a 
wrench is evoked by a twist about T. Hence, T must be a harmonic screw, 
and as there are only n harmonic screws, it is plain that T must coincide 
with H n , and that therefore H n is a conjugate screw of inertia, as well as a 
conjugate screw of the potential, to each one of the remaining n 1 harmonic 
screws. Similar reasoning will, of course, apply to each of the harmonic 
screws taken in succession. 

105. Equations of Motion. 

We now consider the kinetical problem, which may be thus stated. A 
free or constrained rigid body, which is acted upon by a system of forces, is 
displaced by an initial twist of small amplitude, from a position of equi 
librium. The body also receives an initial twisting motion, with a small 
twist velocity, and is then abandoned to the influence of the forces. It is 
required to ascertain the nature of its subsequent movements. 

Let T represent the kinetic energy of the body, in the position of which 
the co-ordinates, referred to the principal screws of inertia, are #/, . . . n . 
Then we have ( 97) : 



while the potential energy which, as before, we denote by V, is an homo 
geneous function of the second order of the quantities #/, . . . # . 

By the use of Lagrange s method of generalized co-ordinates we are 
enabled to write down at once the n equations of motion in the form : 



Substituting for T we have : 




with (n 1) similar equations. Finally, introducing the expression for V 
( 100), we obtain n linear differential equations of the second order. 

The equations which we require can be otherwise demonstrated as follows. 



105] HARMONIC SCREWS. 97 

Suppose the body to be in motion under the influence of the forces, and that 
at any epoch t the co-ordinates of the twisting motion are 

dt " dt~ " 

when referred to the principal screws of inertia. Let / , ... " be the 
co-ordinates of a wrench which, had it acted upon the body at rest for the 
small time e, would have communicated to the body a twisting motion 
identical with that which the body actually has at the epoch t. The 
co-ordinates of the impulsive wrench which would, in the time e, have pro 
duced from rest the motion which the body actually has at the epoch t + e, 
are : 



- 

1 dt "" n dt 

On the other hand, the motion at the epoch t + e may be considered to 
arise from the influence of the wrench / , . . . f M " for the time e, followed by 
the influence of the evoked wrench for the time e. The final effect of the 
two wrenches must, by the second law of motion, be the same as if they 
acted simultaneously for the time e upon the body initially at rest. 

The co-ordinates of the evoked wrench being : 



I dV + J^ 

2p l ddi 2p n ddn 

we therefore have the equation : 

dt 2/>i 
or 

dt 1 dV 



dt ~ 2 Pl d6J 
and w1 similar equations ; but we see from 97 that 

ef -M^W 
e ^- ^^ ~dt 

Differentiating this equation with respect to the time, and regarding e as 
constant, we have 



dt Pl dt* 
whence 

2 <9/ dV 



the same equation as that already found by Lagrange s method. 
B. 



98 THE THEORY OF SCREWS. [105, 

To integrate the equations we assume 

tf^/n, ...#,/=/ n ; 

where f lt ...f n are certain constants, which will be determined, and where H 
is an unknown function of the time : introducing also the value of V, given 
in 100, we find for the equations of motion : 

- MI*/I fj + (4,1/1 + 4 M / + + 4i/) n = o, 

&c. 

c? 2 n 

- Mu n % ^ + (A m f, + An.fi + ...+ Annfn) 11 = 0. 

If the quantity s, and the ratios of the n quantities f 1} .../, be deter 
mined by the n equations : 



(A u + Mu*s-) +f*A lz + ... +f n A m = 0, 
&c., &c. 

+f,A nn _ +... +f n (A nn + Mu^ff) = J 

then the n equations of motion will reduce to the single equation : 



By eliminating /j, ... f n from the n equations, we obtain precisely the 
same equation for s 2 as that which arose ( 104) in the determination of the 
n harmonic screws. The values of fj, ... f n , which correspond to any value 
of s 2 , are therefore proportional to the co-ordinates of a harmonic screw. 

The equation for Q gives : 

H = H sin (st + c). 

Let H l} ... H n , Cj , ... c n be 2n arbitrary constants. Let f pq denote the 
value of/ 9 , when the root s p z has been substituted in the linear equations. 
Then by the known theory of linear differential equations*, 

ff n sin (s n t + c n ), 



0n =fmffi Si" (*i* + C,) + . +fnnH n Sin (s n t + C M ). 

In proof of this solution it is sufficient to observe, that the values of 
#/,... n satisfy the given differential equations of motion, while they also 
contain the requisite number of arbitrary constants. 

* Lagrange s Method, Routh, Rigid Dynamics, Vol. i., p. 369. 



106] HARMONIC SCREWS. 99 

106. Discussion of the Results. 

For the position of the body before its displacement to have been one of 
stable equilibrium, it is manifest that the co-ordinates must not increase 
indefinitely with the time, and therefore all the values of s 2 must be essen 
tially positive, since otherwise the values of #/, .. n would contain expo 
nential terms. 

The 2n arbitrary constants are to be determined by the initial circum 
stances. The initial displacement is to be resolved into n twists about the n 
screws of reference ( 95). This will provide n equations, by making t = 0, 
and substituting for Oi,...0 n , in the equations just mentioned, the amplitudes 
of the initial twists. The initial twisting motion is also to be resolved into 
twisting motions about the n screws of reference. The twist velocities of 

JQ I JQ > 

these components will be the values of ; , . . . , ~, , when t = ; whence 

we have n more equations to complete the determination of the arbitrary 
constants. 

If the initial circumstances be such that the constants H.,, ..., H n are all 
zero, then the equations assume a simple form : 

0i =f u #! sin (s^ + c), 



O n =/!?! sin (sj + c). 

The interpretation of this result is very remarkable. We see that the 
co-ordinates of the body are always proportional to f u , ...,f ln , hence the 
body can always be brought from the initial position to the position at any 
time by twisting it about that screw, whose co-ordinates are proportional to 
fu> >f-m\ but, as we have already pointed out, the screw thus defined is 
a harmonic screw, and hence we have the following theorem : 

If a rigid body occupy a position of stable equilibrium under the 
influence of a conservative system of forces, then n harmonic screws can be 
selected from the screw system of the nth order, which defines the freedom 
of the body, and if the body be displaced from its position of equilibrium 
by a twist about a harmonic screw, and if it also receive any small initial 
twist velocity about the same screw, then the body will continue 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 following 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 

7-2 



100 THE THEORY OF SCREWS. [106-109 

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 constructed, 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 
respectively to the initial amplitude of the twist, and to the twist velocity, 
which have been assigned to the corresponding harmonic screw. 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 equilibrium about the harmonic 
screws, whose amplitudes are equal to these arcs. 

107. Remark 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 depart 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 rj, which is evoked by the twist. But the actual effect of an 
impulsive wrench on 77 would be to make the body twist about the harmonic 
screw ( 104), and as the continued action of the wrench on 77 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 X. 

FREEDOM OF THE FIRST ORDER. 

108. Introduction. 

In the present chapter we shall apply the principles developed in the 
preceding chapters to study the Statics and Dynamics of a rigid body 
which has freedom of the first order. 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 special screw system which defines the freedom enjoyed by 
the body. We shall then be enabled to determine the reciprocal screw 
system, 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 any screw of the screw system. 
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 of 
forces, the movements of the body being limited as before to the screws of 
the screw system. 

109. Screw System 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 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 system consists of one screw, whose pitch is zero ; 
in the latter case the screw system consists of one screw, whose pitch is 
infinite. 



102 THE THEORY OF SCREWS. [HO, 

110. The Reciprocal Screw System. 

The integer which denotes the order of a screw system, and the integer 
which denotes the order of the reciprocal screw system, will, in all cases, 
have the number six for their sum ( 72). Hence a screw system of the 
first order will have as its reciprocal a screw system of the fifth order. 

For a screw 6 to belong to a screw system of the fifth order, the necessary 
and sufficient condition is, that 6 be reciprocal to one given screw a. This 
condition is expressed in the usual form : 

(p a + Pe) cos d ag sin = 0, 

where is the angle, and d ae the perpendicular distance between the screws 
9 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 system of the fifth 
order. For the straight line and a being given, d a6 and are determined, 
and hence the pitch p e can be determined by the linear equation just 
written. 

Consider next a point A, and the screw a. Every straight line through 
A, when furnished with the proper pitch, will be reciprocal to a. Since the 
number of lines through A is doubly infinite, it follows that a singly infinite 
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, 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*, arid then we shall 
deduce the more general theorem. 

By a twist of small amplitude about a the point A is moved to an adja 
cent point B. To effect this movement against a force at A which is per 
pendicular to AB, no work will be required; hence every line through A, 
perpendicular to AB, may be regarded as a screw of zero pitch, reciprocal 
to a. 

We must now enunciate a principle which applies to a screw system of 
any order. We have already referred to it with respect to the cylindroid 
( 18). If all the screws of a screw system be modified by the addition of 
the same linear magnitude (positive or negative) to the pitch of every screw, 
then the collection of screws thus modified still form a screw system 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 system be 

* This theorem is due to Mdbius, 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 Zusammensetzung unendlich kleiner Drehungen," Crelle s 
Journal, t. xviii., pp. 189212.) (Berlin, 1838.) 



HI] FREEDOM OF THE FIRST ORDER. 103 

increased by a certain quantity, and all the pitches of the reciprocal system 
be diminished by the same quantity, then all the first set of screws thus 
modified are reciprocal to all the second group as modified. Hence, since a 
screw-system of the nth order consists of all the screws reciprocal to 6 n 
screws, it follows that the modified set must still be a screw system. 

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 ij, of pitch p a + k, on the same line as or, then we have 
just shown that all the screws p, of zero pitch, Avhich can be drawn through 
the point A, so as to be reciprocal to ?;, lie in a plane. Since fj, and 77 are 
reciprocal, the screws on the same straight lines as /u, and 77 will be reciprocal, 
provided the sum of their pitches is the pitch of r) ; therefore, a screw X, of 
pitch k, on the same straight line as p, will be reciprocal to the screw a, of 
pitch p a ; but all the lines //. 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 1} & 3 , ..., will 
correspond a point A 1} A. ... ; and it is worthy of remark, that all the points 
A l} A. 2 must lie on a right line which intersects a at right angles; for join 
A l} A, then a screw on the line A^.^, which has for pitch either k : or k., 
must be reciprocal to a; but this is impossible unless A^A^ intersect a at 
a right angle. 

111. 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 system 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. 

If two wrenches act upon the body, then the condition of equilibrium is, 
that, when the two wrenches are compounded by the aid of a cylindroid, 
the single wrench which replaces them shall lie upon that one screw of the 
cylindroid, which is reciprocal to a ( 2G). 

We can express with great facility, by the aid of screw co-ordinates, the 
condition that wrenches of intensities 6", <", on two screws 6, <f>, shall 
equilibrate, when applied to a body only free to twist about a. 

Adopting any six co-reciprocals as screws of reference, and resolving each 
of the wrenches on 9 and </> into its six components on the six screws of 



104 THE THEORY OF SCREWS. [ill- 

reference, we shall have for the intensity of the component of the resultant 
wrench on &> n 

6" 6 n + 4>"<j> n - 

Hence the co-ordinates of the resultant wrench are proportional to 



For equilibrium this screw must be reciprocal to a, whence we have 
Pl a, (ff 0, + f >0 +...+P*. (ff Ot + f &) = 0, 

Or, 0"^a6 + $"1*04 = 0. 

This equation merely expresses that the sum of the works done in a 
small twist about a against the wrenches on and <f> is zero. 

We also perceive that a given wrench may be always replaced by a 
wrench of appropriate intensity on any other screw, in so far as the effect 
on a body only free to twist about a is concerned. 

It may not be out of place to notice the analogy which the equation just 
written bears to the simple problem of the determination of the condition 
that two forces should be unable to disturb the equilibrium of a particle 
only free to move on a straight line. If P, Q be the two forces, and if I, m 
be the angles which the forces make with the direction in which the particle 
can move, then the condition is 

P cos I + Q cos in = 0. 

This suggests an analogy between the virtual co-efficient of two screws, 
and the cosine of the angle between two lines. 

112. Particular Case. 

If a body having freedom of the first order be in equilibrium under the 
action of gravity, then the vertical through the centre of inertia must lie 
in the plane of reciprocal screws of zero pitch, drawn through the centre 
of inertia. 

113. Impulsive Forces. 

If an impulsive wrench of intensity 77 " act on the screw >/, while the 
body is only permitted to twist about a, then we have seen in 90 how the 
twist velocity produced can be found. We shall now determine the impulsive 
reaction of the constraints. This reaction must be an impulsive wrench of 
intensity X " on a screw X, which is reciprocal to a. The determination of X 
may be effected geometrically in the following manner : Let /* be the screw, 
an impulsive wrench on which would, if the body were perfectly free, cause 
an instantaneous twisting motion about a ( 80). Draw the cylindroid (77, fi). 



114] 



FREEDOM OF THE FIRST ORDER. 



105 



Then A, must be that screw on the cylindroid which is reciprocal to a, for a 
wrench on X, and the given wrench on 77, must compound into a wrench on p, 
whence the three screws must be co-cylindroidal * ; also X must be reciprocal 
to a, so that its position on the cylindroid is known ( 20). Finally, as the 
impulsive intensity 17 " is given, and as the three screws 77, X, /* are all 
known, the impulsive intensity X " becomes determined ( 14). 

114. Small Oscillations. 

We shall now suppose that a rigid body which has freedom of the first 
order occupies a position of stable equilibrium under the influence of a 
system of forces. If the body be displaced by a small twist about the screw 
a which prescribes the freedom, and if it further receive a small initial twist 
velocity about the same screw, the body will continue to perform small 
twist oscillations about the screw a. We propose to determine the time 
of an oscillation. 



The kinetic energy of the body, when animated by a twist velocity 



7 / 

, 

CLt> 

is Mu a ~ (~ J ( 89). The potential energy due to the position attained by 

giving the body a twist of amplitude a away from its position of equili 
brium, is Fv a 2 af 2 ( 102). But the sum of the potential and kinetic energies 
must be constant, whence 



Mu a 



-?-) 
dt I 



Fv a -a- = const. 



Differentiating we have 



Integrating this equation we have 

a = A sin A/ 






- 
Mu a 



t + B cos 



/ Fv a 
V Mu 



where A and B are arbitrary constants. The time of one oscillation is 
therefore 

IM 
F" 

Regarding the rigid body and the forces as given, and comparing 
inter se the periods about different screws a, on which the body might have 
been constrained to twist, we see from the result just arrived at that the 

time for each screw a is proportional to -- . 



We shall often for convenience speak of three screws on the same cylindroid as co-cylindroidal. 



106 THE THEORY OF SCREWS. [115-117 

115. Property of Harmonic Screws. 

As the time of vibration is affected by the position of the screw to which 
the motion is limited, it becomes of interest to consider how a screw is to 
be chosen so that the time of vibration shall be a maximum or minimum. 
With slightly increased generality we may state the problem as follows : 

Given the potential for every position in the neighbourhood of a position 
of stable equilibrium, it is required to select from a given screw system 
the screw or screws on which, if the body be constrained to twist, the time 
of vibration will be a maximum or minimum, relatively to the time of 
vibration on the neighbouring screws of the same screw system. 

Take the n principal screws of inertia belonging to the screw system, 
as screws of reference, then we have to determine the n co-ordinates of a 

screw a by the condition that the function shall be a maximum or a 

V a 

minimum. 

Introducing the value of u a ( 97), and of v a ( 102), in terms of the 
co-ordinates, we have to determine the maximum and minimum of the 
function 



Multiplying this equation by the denominator of the left-hand side, 
differentiating with respect to each co-ordinate successively, and observing 
that the differential coefficients of a; must be zero, we have the n equations : 

(^ n - U^X) j + A 12 dz ... + A ln n = 0, 

&c., &c. 

AM&! + A H .,a., ... -I- (A nn ~ Unty H = 0. 

We hence see that there are n screws belonging to each screw of the 
nth order on which the time of vibration is a maximum or minimum, and by 
comparison with 104 we deduce the interesting result that these n screws 
are also the harmonic screws. 

Taking the screw system of the sixth order, which of course includes 
every screw in space, we see that if the body be permitted to twist about 
one of the six harmonic screws the time of vibration will be a maximum 
or minimum, as compared with the time of vibration on any adjacent screw. 

If the six harmonic screws were taken as the screws of reference, then 
u a 2 and w a 2 would each consist of the sum of six square terms ( 89, 102). If 
the coefficients in these two expressions were proportional, so that u a " only 
differed from v^ by a numerical factor, we should then find that every screw 
in space was an harmonic screw, and that the times of vibrations about 
all these screws were equal. 



CHAPTER XL 



FREEDOM OF THE SECOND ORDER. 

116. The Screw System of the Second Order. 

When a rigid body is capable of being twisted about two screws and 
(j), it is capable of being twisted about every screw on the cylindroid (6, <) 
( 14). If it also appear that the body cannot be twisted about any screw 
which does not lie on the cylindroid, then as we know the body has freedom 
of the second order, and the cylindroid is the screw system of the second 
order by which the freedom is defined ( 219). 

Eight numerical data are required for determination of a cylindroid (75). 
We must have four for the specification of the nodal line, two more are 
required to define the extreme points in which the surface cuts the nodal 
line, one to assign the direction of one generator, and one to give the pitch 
of one screw, or the eccentricity of the pitch conic. 

Although only eight constants are required to define the cylindroid, yet 
ten constants must be used in defining two screws 6, (f>, from which the 
cylindroid could be constructed. The ten constants not only define tne 
cylindroid, but also point out two special screws upon the surface ( 77). 

117. Applications of Screw Co-ordinates. 

We have shown ( 40) that if a, /3 be the two screws of a cylindroid, 
which intersect at right angles, then the co-ordinates of any screw 9, which 
makes an angle I with the screw a, are : 

ttj cos I + & sin I, . . . a s cos I + /9 (j sin I, 
reference being made as usual to any set of six co-reciprocals. 

In addition to the examples of the use of these co-ordinates already 
given ( 40), we may apply them to the determination of that single screw 
6 upon the cylindroid (a, /8), which is reciprocal to a given screw rj. 



108 THE THEORY OF SCREWS. [117- 

From the condition of reciprocity we must have : 

P\n\ (i cos I + & sin /)+...+ p ti rj 6 ( cos I + /3 6 sin I) = 0, 
or, OT ar| cos I + zap,, sin I = 0. 

From this tan I is deduced, and therefore the screw becomes known 
( 26). 

In general if tn^ be the virtual coefficient of any screw 77 and a screw 
on the cylindroid, we have 

V*r,0 = ^ar, COS I + CT^, SU1 I \ 

whence if on each screw a distance be set off from the nodal line equal to 
the virtual coefficient between 77 and 6, the points thus found will lie on 
a right circular cylinder, of which the equation is ; 

X? + y* = TS^X + TX^y. 

Thus the screw which has the greatest virtual coefficient with 77 is at 
right angles to the screw reciprocal to 77, and in general two screws can be 
found upon the cylindroid which have a given virtual coefficient with any 
given external screw. 

118. Relation between Two Cylindroids. 

We may here notice a curious reciprocal relation between two cylindroids, 
which is manifested when one condition is satisfied. If a screw can be found 
on one cylindroid, which is reciprocal to a second cylindroid, then conversely 
a screw can be found on the latter, which is reciprocal to the former. Let 
the cylindroids be (a, /3), and ( X, fi). If a screw can be found on the former, 
which is reciprocal to the latter, then we have : 

pAi (i cos I + /3j sin 1) -f . . . + p n ^n (oi cos I + /3 n sin I) = 0, 
Pif^i (i cos I + ft sin 1) + . . . 4- pn^n ( a n cos I + ft n sin I) = 0. 
Whence eliminating I, we find : 



As this relation is symmetrical with regard to the two cylindroids, the 
theorem has been proved. 

119. Co-ordinates of Three Screws on a Cylindroid. 

The co-ordinates of three screws upon a cylindroid are connected by four 
independent relations. In fact, two screws define the cylindroid, and the 
third screw must then satisfy four equations of the form ( 20). These 
relations can be expressed most symmetrically in the form of six equations, 
which also involve three other quantities. 



120] 



FREEDOM OF THE SECOND ORDER. 



109 



Let X, JJL, v be three screws upon a cylindroid, and let A, B, C denote the 
angles between /LI v, between v X, and between X //., respectively. If wrenches 
of intensities X", // , v", on \, /*, v, respectively, are in equilibrium, we must 
have ( 14): 

_X" p" v" 

sin A sin B ~~ sin C 

But we have also as a necessary condition that if each wrench be resolved 
into six component wrenches on six screws of reference, the sum of the 
intensities of the three components on each screw of reference is zero ; 
whence 

Xj sin A + /ij sin B + v l sin (7 = 0, 



X sin A + p, 6 sin B + v 6 sin G = 0. 
From these equations we deduce the following corollaries : 

The screw of which the co-ordinates are proportional to aXj + bfa , ... 
X 6 + bfj, G , lies on the cylindroid (X, //.), and makes angles with the screws 
X, IJL, of which the sines are inversely proportional to a and 6. 

The two screws, of which the co-ordinates are proportional to 
aXj 6/ij, ... aX 6 + &/A G , 

and the two screws X, /* are respectively parallel to the four rays of a plane 
harmonic pencil. 

120. 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 9, <f>, about which 
the body can be twisted, happen to lie on the same straight line, then the 
cylindroid becomes illusory. If the amplitudes of the two twists be 6 , $> , 
then the body will have received a rotation 6 + <f> , accompanied by a trans 
lation p e + <f> p<t>. This movement is really identical with a twist on a 
screw of which the pitch is : 

O pe + 



& + <!> 

Since , </> may have any ratio, we see that, under these circumstances, the 
screw system which defines the freedom consists of all the screws with 
pitches ranging from -co to +00, which lie along the given line. It 
follows ( 47), that the co-ordinates of all the screws about which the 
body can be twisted are to be found by giving x all the values from 
QO to + GO in the expressions : 



a x dR * x dR 
l + ~ " K + 



in which 



L10 THE THEORY OF SCREWS. [121- 

121. Displacement of a Point. 

Let P be a point, and let a, /3 be any two screws upon a cylindroid. If 
a body to which P is attached receive a small twist about a, the point P will 
be moved to P . If the body receive a small twist about /3, the point P 
would be moved to P". Then whatever be the screw 7 on the cylindroid 
about which the body be twisted, the point P will still be displaced in the 
plane PP P". 

For the twist about 7 can be resolved into two twists about a and /3, and 
therefore every displacement of P must be capable of being resolved along 
PP and PP". 

Thus through every point P in space a plane can be drawn to which the 
small movements of P, arising from twists about the screws on a given 
cylindroid are confined. The simplest construction for this plane is as 
follows: Draw through the point P two planes, each containing one of the 
screws of zero pitch; the intersection of these planes is normal to the 
required plane through P. 

The construction just given would fail if P lay upon one of the screws 
of zero pitch. The movements of P must then be limited, not to a plane, 
but to a line. The line is found by drawing a normal to the plane passing 
through P, and through the other screw of zero pitch. 

We thus have the following curious property due to M. Mannheim*, viz., 
that a point in the rigid body on the line of zero pitch will commence to 
move in the same direction whatever be the screw on the cylindroid about 
which the twist is imparted. 

This easily appears otherwise. Appropriate twists about any two screws, 
a and /3, can compound into a twist about the screw of zero pitch X, but the 
twist about X cannot disturb a point on X. Therefore a twist about ft must 
be capable of moving a point originally on X back to its position before it 
was disturbed by a. Therefore the twists about /3 and a. must move the 
point in the same direction. 

122. Properties of the Pitch Conic. 

Since the pitch of a screw on a cylindroid is proportional to the inverse 
square of the parallel diameter of the pitch conic ( 18), the asymptotes 
must be parallel to the screws of zero pitch ; also since a pair of reciprocal 
screws are parallel to a pair of conjugate diameters ( 40), it follows that 
the two screws of zero pitch, and any pair of reciprocal screws, are parallel 
to the rays of an harmonic pencil. If the pitch conic be an ellipse, there 

* Journal de I ecole Polytechnique, T. xx. cah. 43, pp. 57122 (1870). 



123] FREEDOM OF THE SECOND ORDER. Ill 

are no real screws of zero pitch. If the pitch conic be a parabola, there is 
but one screw of zero pitch, and this must be one of the two screws which 
intersect at right angles. 

123. Equilibrium of a Body with Freedom of the Second Order. 

We shall now consider more fully the conditions under which a body 
which has freedom of the second order is in equilibrium. The necessary 
and sufficient condition is, that the forces which act upon the body shall 
constitute a wrench upon a screw which is reciprocal to the cylindroid which 
defines the freedom of the body. 

It has been shown ( 23), that the screws which are reciprocal to a cylin 
droid exist in such profusion, that through every point in space a cone of 
the second order can be drawn, of which the entire superficies is made up of 
such screws. We shall now examine the distribution of pitch upon such a 
cone. 

The pitch of each reciprocal screw is equal in magnitude, and opposite in 
sign, to the pitches of the two screws of equal pitch, in which it intersects the 
cylindroid ( 22). Now, the greatest and least pitches of the screws on the 
cylindroid are p a and p$ ( 18). For the quantity p a cos 2 1 + p ft sin 2 1 is always 
intermediate between p a cos 2 1 + p a sin 2 1 and pp cos 2 1 + pp sin 2 1. Hence it 
follows that the generators of the cone which meet the cylindroid in three 
real points must have pitches intermediate between p a and pp. It is also 
to be observed that, as only one line can be drawn through the vertex of 
the cone to intersect any two given screws on the cylindroid, so only one 
screw of any given pitch can be found on the reciprocal cone. 

One screw can be found upon the reciprocal cone of every pitch from 
oo to + oo . The line drawn through the vertex parallel to the nodal line 
is a generator of the cone to which infinite pitch must be assigned. Setting 
out from this line around the cone the pitch gradually decreases to zero, 
then becomes negative, and increases to negative infinity, when we reach 
the line from which we started. We may here notice that when a screw 
has infinite pitch, we may regard the infinity as either + or indifferently. 
If we conceive distances marked upon each generator of the cone from the 
vertex, equal to the pitch of that generator, then the parallel to the nodal 
line drawn from the vertex forms an asymptote to the curve so traced upon 
the cone. It is manifest that we must admit the cylindroid to possess 
imaginary screws, whose pitch is nevertheless real. 

The reciprocal cone drawn from a point to a cylindroid, is decomposed 
into two planes, when the point lies upon the cylindroid. The first plane 
is normal to the generator passing through the point. Every line in this 
plane must, when it receives the proper pitch, be a reciprocal screw. The 



112 THE THEORY OF SCREWS. [123- 

second plane is that drawn through the point, and through the other screw 
on the cylindroid, of equal pitch to that which passes through the point. 

We have, therefore, solved in the most general manner the problem of 
the equilibrium of a rigid body with two degrees of freedom. We have 
shown that the necessary and sufficient condition is, that the resultant 
wrench be about a screw reciprocal to the cylindroid expressing the freedom, 
and we have seen the manner in which the reciprocal screws are distributed 
through space. We now add a few particular cases. 

124. Particular Cases. 

A body which has two degrees of freedom is in equilibrium under the 
action of a force, whenever the line of action of the force intersects both 
the screws of zero pitch upon the cylindroid. 

If a body acted upon by gravity have freedom of the second order, the 
necessary and sufficient condition of equilibrium is, that the vertical through 
the centre of inertia shall intersect both of the screws of zero pitch. 

A body which has freedom of the second order will be in equilibrium, 
notwithstanding the action of a couple, provided the axis of the couple be 
parallel to the nodal line of the cylindroid. 

A body which has freedom of the second order will remain in equilibrium, 
notwithstanding the action of a wrench about a screw of any pitch on the 
nodal line of the cylindroid. 

125. The Impulsive Cylindroid and the Instantaneous Cylin 
droid. 

A rigid body M is at rest in a position P, from which it is either partially 
or entirely free to move. If M receive an impulsive wrench about a screw 
X lt it will commence to twist about an instantaneous screw A 1} if, however, 
the impulsive wrench had been about X 2 or X 3 (M being in either case at 
rest in the position P) the instantaneous screw would have been A 2 , or A 3 . 
Then we have the following theorem: 

If Xi, X z , X 3 lie upon a cylindroid S (which we may call the impulsive 
cylindroid), then A 1} A.,, A 3 lie on a cylindroid S (which we may call the 
instantaneous cylindroid). 

For if the three wrenches have suitable intensities they may equilibrate, 
since they are co-cylindroidal ; when this is the case the three instantaneous 
twist velocities must, of course, neutralise; but this is only possible if the 
instantaneous screws be co-cylindroidal ( 93). 



12 5] FREEDOM OF THE SECOND ORDER. 113 

If we draw a pencil of four lines through a point parallel to four gene 
rators of a cylindroid, the lines forming the pencil will lie in a plane. We 
may define the (inharmonic ratio of four generators on a cylindroid to be 
the anharmonic ratio of the parallel pencil. We shall now prove the follow 
ing theorem : 

The anharmonic ratio of four screws on the impulsive cylindroid is equal 
to the anharmonic ratio of the four corresponding screws on the instantaneous 
cylindroid. 

Before commencing the proof we remark that, 

If an impulsive wrench of intensity F acting on the screw X be capable 
of producing the unit of twist velocity about A, then an impulsive wrench 
of intensity Fta on X will produce a twist velocity w about A. 

Let X 1} X 2 , X 3) X^ be four screws on the impulsive cylindroid, the 
intensities of the wrenches appropriate to which are F^, F 2 a) 2 , F 3 to 3t Fw. 
Let the four corresponding instantaneous screws be A 1} A 2 , A s , A^ and the 
twist velocities be a> 1; &> 2 , a> 3 , a> 4 . Let <j) m be the angle on the impulsive 
cylindroid defining X m , and let 6 m be the angle on the instantaneous 
cylindroid defining A m . 

If three impulsive wrenches equilibrate, the corresponding twist velocities 
neutralize by the second law of motion : hence ( 14) certain values of 
&>!, <B.;,, 6) :! , ft) 4 must satisfy the following equations: 



to, 



sin (0 2 - 3 ) sin (0 3 - 0,) sin (0, - a ) 



sin (fa (j> 3 ) sin (</> 3 fa) sin (fa 



too 



sin (0 3 - 4 ) sin (0 4 - 2 ) sin (0 2 - :! ) 



_ 

sin (0 :j - fa) ~ sin (fa - fa) sin (fa - fa*) 
whence 

sin (0j 0) sin (0 3 4 ) _ sin (fa fa) sin (<ft :i fa) 
sin (0 3 0!> sin (0 4 - 2 ) ~ sin (fa fa) sin (fa fa,) 

which proves the theorem. 

If we are given three screws on the impulsive cylindroid, and the 
corresponding three screws on the instantaneous cylindroid, the connexion 
between every other corresponding pair is, therefore, geometrically deter 
mined. 

B. $ 



114 THE THEORY OF SCREWS. [126- 

126. Reaction of Constraints. 

Whatever the constraints may be, their reaction produces an impulsive 
wrench R l upon the body at the moment of action of the impulsive wrench 
X^. The two wrenches X l and 7^ compound into a third wrench F,. If 
the body were free, Y l is the impulsive wrench to which the instantaneous 
screw A-i would correspond. Since X lt X 2 , X 3 are co-cylindroidal, A lt A z , A 3 
must be co-cylindroidal, and therefore also must be Y lt F 2 , Y 3 . The nine 
wrenches X ly X. 2 , X 3 , R lt E, R i} - F,, F 2 , Y 3 must equilibrate; but if 
Xi, X 2 , X 3 equilibrate, then the twist velocities about A lt A 2 , A a must 
neutralize, and therefore the wrenches about F, , F 2 , F 3 must equilibrate. 
Hence RI, R. 2 , R 3 equilibrate, and are therefore co-cylindroidal. 

Following the same line of proof used in the last section, we can show 
that 

If impulsive wrenches on any four co-cylindroidal screws act upon a 
partially free rigid body, the four corresponding initial reactions of the 
constraints also constitute wrenches about four co-cylindroidal screws; and, 
further, the anharmonic ratios of the two groups of four screws are equal. 

127. Principal Screws of Inertia. 

If a quiescent body with freedom of the second order receive impulsive 
wrenches on three screws X l} X. 2 , X 3 on the cylindroid which expresses the 
freedom, and if the corresponding instantaneous screws on the same cylin 
droid be A lt A%, A s , 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 lt X 2 , X 3 is equal 
to the anharmonic ratio of A, A^, A.,, A 3 . 

If three rays parallel to X lt X 2 , X 3 be drawn from a point, and from the 
same point three rays parallel to A^, A.,, A 3 , then, all six rays being in the 
same plane, it is well known that the problem to determine a ray Z such 
that the anharmonic ratio of Z, A l , A 2 , A 3 is equal to that of Z, X^, X 2 , X 3) 
admits of two solutions. There are, therefore, two screws on a cylindroid 
such that an impulsive wrench on one of these screws will cause the 
body to commence to twist about the same screw. 

We have thus arrived by a special process at the two principal screws of 
inertia possessed by a body which has freedom of the second order. This is, 
of course, a particular case of the general theorem of 78. We shall show 
in the next section how these screws can be determined in another manner. 

128. The Ellipse of Inertia. 

We have seen ( 89) that a linear parameter u a may be conceived appro 
priate to any screw a of a system, so that when the body is twisting about 



128] FREEDOM OF THE SECOND ORDER. 115 

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 a . 

We are now going to consider the distribution of this magnitude u a on 
the screws of a cylindroid. If we denote by u lt u^ the values of ti a for any 
pair of conjugate screws of inertia on the cylindroid (81), and if by a l , a, 
we denote the intensities of the components on the two conjugate screws of 
a wrench of unit intensity on a, we have ( 97) 



From the centre of the cylindroid draw two straight lines parallel to the 
pair of conjugate screws of inertia, and with these lines as axes of as and y 
construct the ellipse of which the equation is 

U-fX* + M 2 2 7/ 2 = H, 

where H is any constant. If r be the radius vector in this ellipse, we have 
( 35) 

x y 

- = ! and - = a,, ; 

r r 

whence by substitution we deduce 



which proves the following theorem: 

The linear parameter u a on any screw of the cylindroid is inversely 
proportional to the parallel diameter of a certain ellipse, and a pair of 
conjugate screws of inertia on the cylindroid are parallel to a pair of 
conjugate diameters of the same ellipse. This ellipse may be called the 
ellipse of inertia. 

The major and minor axes of the ellipse of inertia are parallel to screws 
upon the cylindroid, which for a given twist velocity correspond respectively to 
a maximum and minimum kinetic energy. 

An impulsive wrench on a screw 77 acts upon a quiescent rigid body 
which has freedom of the second order. It is required to determine the 
screw 6 on the cylindroid expressing the freedom about which the body 
will commence to twist. 

The ellipse of inertia enables us to solve this problem with great facility. 
Determine that one screw <f> on the cylindroid which is reciprocal to ?? ( 26). 
Draw a diameter D of the ellipse of inertia parallel to <. Then the required 
screw 6 is simply that screw on the cylindroid which is parallel to the 
diameter conjugate to D in the ellipse of inertia. 

The converse problem, viz., to determine the screw 77, an impulsive wrench 

82 



116 THE THEORY OF SCREWS. [128, 

on which would make the body commence to twist about 6, is indeterminate. 
Any screw in space which is reciprocal to </> would fulfil the required condition 
(136). 

We have seen in 96 that an impulsive wrench on any screw in space may 
generally be replaced by a precisely equivalent wrench upon the cylindroid 
which expresses the freedom. We are now going to determine the screw 77, 
on the cylindroid of freedom, an impulsive wrench on which would make the 
body twist about a given screw 6 on the same cylindroid. This can be easily 
determined with the help of the pitch conic ; for we have seen ( 40) that a 
pair of reciprocal screws on the cylindroid of freedom are parallel to a pair 
of conjugate diameters of the pitch conic. The construction is therefore as 
follows: Find the diameter A which is conjugate, with respect to the ellipse 
of inertia, to the diameter parallel to the given screw 6. Next find the 
diameter B which is conjugate to the diameter A with respect to the pitcfi 
conic. The screw on the cylindroid parallel to the line B thus determined 
is the required screw 77. 

Two concentric ellipses have one pair of common conjugate diameters. 
In fact, the four points of intersection form a parallelogram, to the sides of 
which the pair of common conjugate diameters are parallel. We can now 
interpret physically the common conjugate diameters of the pitch conic, and 
the ellipse of inertia. The two screws on the cylindroid parallel to these 
diameters are conjugate screws of inertia, and they are also reciprocal; they 
are, therefore, the principal screws of inertia, to which we have been already 
conducted (127). 

If the distribution of the material of the body bear certain relations to 
the arrangement of the constraints, we can easily conceive that the pitch 
conic and the ellipse of inertia might be both similar and similarly situated. 
Under these exceptional circumstances it appears that every screw of the 
cylindroid would possess the property of a principal screw of inertia. 

129. The Ellipse of the Potential. 

We are now to consider another ellipse, which, though possessing many 
useful mathematical analogies to the ellipse of inertia, is yet widely different 
from a physical point of view. We have introduced ( 102) the conception 
of the linear magnitude w , the square of which is proportional to the work 
done in effecting a twist of given amplitude about a screw a. from a position 
of stable equilibrium under the influence of a system of forces. We now 
propose to consider the distribution of the parameter v a upon the screws of 
a cylindroid. It appears from 102 that if v l} v denote the values of the 
quantity v a for each of two conjugate screws of the potential, and if a,, 2 
denote the intensities of the components on the two conjugate screws of a 



129] FREEDOM OF THE SECOND ORDER. 117 

wrench of unit intensity on a screw a, which also lies upon the cylindroid, 
then 

* = 



From the centre of the cylindroid draw two straight lines parallel to the 
pair of conjugate screws of the potential, and with these lines as axes of x 
and y construct the ellipse, of which the equation is 

vi*a? + v./y* = H, 
where H is any constant. If r be the radius vector in this ellipse, we have 

x y 

= a, and - = a., ; 
r r 

whence by substitution we deduce 

-- i ^;* a :; 

which proves the following theorem : 

The linear parameter v a on any screw of the cylindroid is inversely pro 
portional to the parallel diameter of a certain ellipse, and a pair of conjugate 
screws of the potential are parallel to a pair of conjugate diameters of the 
same ellipse. 

This ellipse may be called the ellipse of the potential. 

The major and minor axes of the ellipse of the potential are parallel to 
screws upon the cylindroid, which, for a twist of given amplitude, correspond 
to a maximum and minimum potential energy. 

When the body has to relinquish its original position of equilibrium by 
the addition of a wrench on a screw 77 to the forces previously in operation, 
the twist by which the body may proceed to its new position of equilibrium 
is about a screw 0, which can be constructed by the ellipse of the potential. 
Determine the screw </> (on the cylindroid of freedom) which is reciprocal to 
77 ( 26), then </>, and the required screw 0, are parallel to a pair of conjugate 
diameters of the ellipse of the potential. 

The common conjugate diameters of the pitch conic, and the ellipse of 
the potential, are parallel to the two screws on the cylindroid, which we 
have designated the principal screws of the potential ( 101). 

When a body is disturbed from its position of equilibrium by a small 
wrench upon a principal screw of the potential, then the body could be moved 
to the new position of equilibrium required in its altered circumstances by a 
small twist about the same screw. 



118 THE THEORY OF SCREWS. [130- 

130. Harmonic Screws. 

The common conjugate diameters of the ellipse of inertia, and the ellipse 
of the potential, are parallel to the two harmonic screws on the cyliudroid 
( 104). This is evident, because the pair of screws thus determined are 
conjugate screws both of inertia and of the potential. 

If the body be displaced by a twist about one of the harmonic screws, 
and be then abandoned to the influence of the forces, the body will continue 
to perform twist oscillations about that screw. 

If the ellipse of inertia, and the ellipse of the potential, be similar, 
and similarly situated, it follows that every screw on the cylindroid will 
be a harmonic screw. 

131. Exceptional Case. 

We have now to consider the modifications which the results we have 
arrived at undergo when the cylindroid becomes illusory in the case con 
sidered ( 120). 

Suppose that and were a pair of conjugate screws of inertia on the 
straight line about which the body was free to rotate and slide independently. 
Then taking the six absolute principal screws of inertia as screws of reference, 
we must have (97) 

ps dR\ / pf dR\ 

+ f j- hi + -:- , - = o, 

% drjj \ 4^j drjj 
where 77 denotes the screw of zero pitch on the same straight line. 
Expanding this equation, and reducing, we find 



This result can be much simplified. By introducing the condition that 
as in 120 

R = (77! + 77,)- + (7/3 + 1J 4 )- + (rj, + 77 6 ) 2 , 

we obtain 



Hence we can prove ( 133) that in this case the product of the pitches of 
two conjugate screws of inertia is equal to minus the square of the radius of 
gyration about the common axis of the screws. 



132] FREEDOM OF THE SECOND ORDER. 119 

132. Reaction of Constraints. 

We shall now consider the following problem : A body which is free to 
twist about all the screws of a cylindroid C receives an impulsive wrench on 
a certain screw 77. It is required to find the screw X, a wrench on which 
constitutes the impulsive reaction of the constraints. Let C represent the 
cylindroid which, if the body were perfectly free, would form the locus of 
those screws, impulsive wrenches on which correspond to all the screws of C 
as instantaneous screws. Since a wrench on 77, and one on \, make the body 
twist about some screw on C, it follows that the cylindroid (77, X) must have 
a screw p in common with C . The wrench on X might be resolved into two, 
one on 77, and the other on p, and the latter might be again resolved into 
two wrenches on any two screws of C . It therefore follows that X must 
belong to the screw system of the third order, which may be defined by 77, 
and by any two screws from C . Take any three screws reciprocal to this 
system, and any two screws on C. We have then five screws to which X is 
reciprocal, arid it is therefore geometrically determined ( 26). 

When X is found, the cylindroid (77, X) can be drawn, and thus p is deter 
mined. The position of p on C will point out the screw on C, about which 
the body will commence to twist, while the position of p on (77, X), and the 
known intensity of the wrench on 77, will determine the intensity of the 
wrench on X. 



CHAPTER XII. 

PLANE REPRESENTATION OF DYNAMICAL PROBLEMS CONCERNING A BODY 
WITH TWO DEGREES OF FREEDOM*. 

133. The Kinetic Energy. 

If a rigid body of mass M twist about a screw 0, with the twist velocity 
0, then the kinetic energy of the body may be written in the form 

Mufffi, 
where u e is a linear magnitude appropriate to the screw 6 ( 89). 

The function uf is the arithmetic mean between the square of the radius 
of gyration and the square of the pitch, for the kinetic energy of the body 
when twisting about 9 is the sum of two parts : one, the kinetic energy of 
the rotation ; the other, of the translation. The energy of the rotation 
is simply 



this being in accordance with the definition of the radius of gyration p e . 
The kinetic energy due to the translation is, of course, 



whence the total kinetic energy is 
and therefore 



134. Body with two Degrees of Freedom. 

The movements are under these circumstances restricted to twists about 
the screws of a cylindroid, and we shall now examine the law of distribution 

* Royal Irish Academy, Cunningham Memoirs, No. 4, p. 19 (1886) ; see also Proceedings of the 
Royal Irish Academy, 2nd Series, Vol. iv. p. 29 (1883). 



133, 134] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 121 

of u e upon the several screws of the cylindroid ( 128). The representative 
circle ( 50) will give a convenient geometrical construction. 

Let 6 l and 0. 2 be the two co-ordinates of 6 relatively to any two screws 
of reference on the cylindroid. Then the components of the twist velocity 
will be 00i and 00.,. The actual velocity of any point of the body will 
necessarily be a linear function of these components. The square of the 
velocity will contain terms in which 3 is multiplied into 0f, 0^0.,, #./, respec 
tively. If, then, by integration we obtain the total kinetic energy, it must 
assume the form 



whence, from the definition of u g 



The three constants, X, /m, v, are the same for all screws on the cylindroid. 
They are determined by the material disposition of the body relatively to 
the cylindroid. 

We have taken the two screws of reference arbitrarily, but this equation 
can receive a remarkable simplification when the two screws of reference 
have been chosen with special appropriateness. 




Fig. 18. 

Let the lengths AX and BX (fig. 18) be denoted by p l and p 2 , and if e 
be the angle subtended by AB, we have from 57, 

X/?! 2 + 2/zp^ + vpJ - u e ~ (ps - Zptfz cos e + p./) = 0. 

Let us now transform this equation from the screws of reference A, B 
to another pair of screws A , B . Let p^, p., be the distances of X from 
A , B , respectively ; then, from Ptolemy s theorem, we have the following 
equations : 

pl .A B = p.;.AA - pl .AB , 

p,.A B = p 2 .A B-p, .BB . 



122 THE THEORY OF SCREWS. [134 

We thus see that p^ and p., are linear functions of /?/ and p. 2 } the several 
coefficients A B , A B, &c., in these two equations being constant. The 
equation for u is thus to be transformed by a linear substitution for p 1 and 
p.,. Of course ti e , being dependent only upon the position of X, is quite 
unaffected by the change of the screws of reference. We can therefore 
apply the well-known principle that the invariant of this binary quantic 
can only differ by a constant factor from the transformed value. The 

invariant is 

(\ - u e -) (v - u e 2 ) - (yu, + / cos e) 2 . 

This must be true for every point X, and therefore for all values of u g -. 
It is necessary that the coefficients of the terms in the expression 

itg 4 sin 2 e UQ" (X + v -f 2/t cos e) + \v p? 
shall be severally proportional to those in the transformed expression 

u e 4 sin 2 e u g - (X + v -f 2// cos e ) + XV // 2 . 
We thus obtain the two equations of condition, 

sin 2 e _ X + v + 2// cos e _ X i/- // 2 
sin 2 e X + v + 2/it cos e Xp /A 2 

The four quantities, X , /A , v , e , may now be chosen arbitrarily, subject to 
these two equations, which are the necessary as well as the sufficient 
conditions. Indeed it is obvious that there must be but two independent 
quantities corresponding to the two positions of A and B . 

We may impose two conditions on the four quantities, and for our present 

purpose we shall make 

X = v ; /u/ = 0. 

The equations of X and e are then 

sin 2 e _ 2X _V* 

sin 2 e X -4- 2/A cos e + v \v fi 2 
and we obtain 

x/= 2 (^-A^ 
X + 2/A cos e + v 

4 (\v - /u 2 ) 



sin - e = sin 1 e 



- s r, 

(X + 2/LA COS 6 + V) 2 

X is thus uniquely determined, and the expression for sin 2 e gives for e four 
values of the type e , + (TT e ). The negative values are meaningless, and 
the two others are coincident, because the arc which subtends e on one side 
subtends IT e on the other. 

There is thus a single pair of screws of reference which permit the 
expression for u e - to be exhibited in the canonical form 



134] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 123 

We are now led to a simple geometrical representation for u e 2 . Let A, B 
(fig. 19) be the two canonical screws of reference. Bisect AB in , then 



= -2AO -+2XO -, 



= 2XY.XO . 




It is obvious that the point must have a critical importance in the 
kinetic theory, and its fundamental property, which has just been proved, is 
expressed in the following theorem : 

If a rigid body be twisting with the unit of twist velocity about any screw 
X on the cylindroid, then its kinetic energy is proportional to the rectangle 
X . X Y, where is a fixed point. 

We are at once reminded of the theorem of 59, in which a similar 
law is found for the distribution of pitch, only in this case another point, 
0, is used instead of the point . Both points, and , are of much 
significance in the representative circle. We can easily prove the following 
theorem, in which we call the polar of the axis of inertia : 

If a rigid body be twisting with the unit of twist velocity about X, then 
its kinetic energy is proportional to the perpendicular distance from X to the 
axis of inertia. 

The geometrical construction for the pitch given in 51 can also be 
applied to determine w fl 2 . This quantity is therefore proportional to the 
perpendicular from on the tangent at X. It thus appears that the 
representative circle gives a graphic illustration of the law of distribu 
tion of u e - around the screws on a cylindroid. 

The axis of inertia cannot cut the representative circle in real points, for 



124 THE THEORY OF SCREWS. [134- 

otherwise we should have at either intersection a twist velocity without any 
kinetic energy. There is no similar restriction to the axis of pitch. We thus 
see that must always lie inside the circle, but that may be in any 
part of the plane. 

135. Conjugate Screws of Inertia. 

We have already made much use of the conception of Conjugate Screws 
of Inertia. We shall here approach the subject in a manner different from 
that previously employed. 

Let a be a screw about which a rigid body is twisting with a twist 
velocity d ; let the body be simultaneously animated by a twist velocity /? 
about a screw /3. These two will compound into a twist velocity about 
some screw 6. If the body only had the first twist velocity, its kinetic 
energy would be Mv-ffi. If it only had the second, the energy would be 
Muf fP. When it has both twist velocities together, the kinetic energy is 
Mufti*. Generally it will not be true that the resulting kinetic energy is 
equal to the sum of the components ; but, under a special relation between 
a and /8, we can have this equality ; and as shown in 88 under these cir 
cumstances a and /3 are conjugate screws of inertia. The necessary condition 
is thus expressed : 

ufu* = u a 2 d* + M/j 2 /3 2 . 

We have now to prove the following important theorem : 

Any chord through the pole of the axis of inertia intersects the representa 
tive circle in a pair of conjugate screws of inertia. 

For we have 

0- tct 2 :^ 2 :: AR- : BX* : AX 2 ; 

but if AB passes through the pole of the axis of inertia, then the centre of 
gravity of masses AIF at X, + BX" at A, and + AX- at B, will lie on the 
axis of inertia ; and, accordingly, 



uf = BX-u a 2 + 
whence 



which proves the theorem. 

Or we might have proceeded thus: From Ptolemy s theorem (fig. 19), 

AB.XY = AX .BY+AY.BX: 
multiplying by AB . XO , 



136] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 

but, from the property of the circle, 

B Y. XO = AX . BO ; A Y . XO = EX . A ; 
whence 

AB .XY. XO =AX\ AB. BO + BX>.AB.AO , 

from which we obtain, as before, 



125 



136. Impulsive Screws and Instantaneous Screws. 

A rigid body having two degrees of freedom lies initially at rest. It is 
suddenly acted upon by an impulsive wrench of large intensity acting for a 
short time. The body will, in general, commence to move by twisting about 
some screw on the cylindroid, and the kinetic problem now to be studied is 
the following : Given the impulsive screw, and the intensity of the impul 
sive wrench, find the instantaneous screw and the acquired twist velocity. 

The problem will be rendered more concise by the conception of the 
reduced wrench ( 96). It is to be remembered, that as the body is only 
partially free, there are an infinite number of screws on which wrenches 
would make the body commence to twist about a given screw on the cylin 
droid. For, let 6 be an impulsive screw situated anywhere, and let an 
impulsive wrench on cause the body to commence to move by twisting 
about some screw, a, on the cylindroid. Let \, /u-, v, p be any four screws 
reciprocal to the cylindroid. Then any wrench on a screw belonging to the 
system defined by these five screws will make the body commence to move 
by twisting about a. Let e be that one screw on the cylindroid which is 
reciprocal to 6, then e is reciprocal to the whole system defined by A,, /*, v, p, 
0, and, conversely, each screw of this system will be reciprocal to e. We 
thus see that any screw, wherever situated, provided only that it is 
reciprocal to e, will be an impulsive screw corresponding to a as an instan 
taneous screw. Any one of this system may, with perfect generality, be 
chosen as the impulsive screw. Among them there is one which has a 
special feature. It is that screw, </>, on the cylindroid which is reciprocal to 
77 ; and hence we have the following theorem ( 128): 

Given any screw, a, on the cylindroid, then there is in general another screw, 
</>, also on the cylindroid, such that an impulsive wrench administered on <f) 
will make the body twist about a. 

This correspondence of the two systems of screws must be of the one-to- 
one type ; for, suppose that two impulsive screws on the cylindroid had the 
same instantaneous screw, it would then be possible for two impulsive 
wrenches, of properly chosen intensities on two different screws, to produce 



126 THE THEORY OF SCREWS. [136- 

equal and opposite twist velocities on the common instantaneous screw. 
The body would then not move, and therefore the two impulsive wrenches 
must equilibrate. But this is impossible, if they are on two different 
screws. 

137. Two Homographic Systems. 

From what has been shown it might be expected that the points corre 
sponding to the instantaneous screws and those corresponding to the 
impulsive screws should, on the representative circle, form two homographic 
systems. That this is so we shall now prove. 




Let A, B (fig. 20) be a pair of impulsive screws, and let A , B be respec 
tively the corresponding pair of instantaneous screws, i.e. an impulsive 
wrench on A will make the body commence to twist about A , and similarly 
for B and B . Let an impulsive wrench on A, of unit intensity, generate a 
twist velocity, a, about A , and let /3 be the similar quantity for B and B . 

Let X be any other screw on which an impulsive wrench is to be applied 
to the body supposed quiescent. The body will commence to twist about 
some other screw, X , with a certain twist velocity fa. We can determine 
<Z> in the following manner: The unit impulsive wrench on X can be 
replaced by two component wrenches on A and B, the intensities of these 
being 

BX A;X 

~AB AB 

respectively. 

These impulsive wrenches will generate about A , B twist velocities 
respectively equal to 

BX 



138] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 127 

those components must, when compounded, produce the twist velocity &> about 
X , and, accordingly, we have 

. BX . B X AX . A X 



Retaining A, B, A , B , as before, let us now introduce a second pair of points, 
Y and F , instead of X and X , and writing &> instead of o>, we have 

.BY .,FT 



_ 
a AB ~ * A B AB ~ A B 

whence, eliminating a, /3, &>, o> , we have 

5Z BY B X FT 
AX : AY :: A X : A Y" 

As the length of a chord is proportional to the sine of the subtended 
angle, we see that the anharmonic ratio of the pencil, subtended by the four 
points A, B, X, Fat a point on the circumference, is equal to that subtended 
by their four correspondents, A , B , X , Y . We thus learn the following 
important theorem : 

A system of points on the representative circle, regarded as impulsive 
screws, and the corresponding system of instantaneous screws, form two homo- 
graphic systems. 

138. The Homographic Axis. 

Let A, B, C, D (fig. 21) represent four impulsive screws, and let A , B , 
C , D be the four corresponding instantaneous screws. Then, by the well- 




Fig. 21. 



known homographic properties of the circle, the three points, L, M, N, will 
be collinear, and we have the following theorem : 



128 THE THEORY OF SCREWS. [138- 

If A and B be any two impulsive screws, and if A and B be the corre 
sponding instantaneous screws, then the chords AR and BA will always 
intersect upon the fixed right line XY. 

This right line is called the homographic axis. It intersects the circle in 
two points, X and Y, which are the double points of the homographic systems. 
These points enjoy a special dynamical significance. They are the two 
Principal Screws of Inertia, and hence 

The homographic axis intersects the circle in two points, each of which 
possesses the property, that an impulsive wrench administered on that screw will 
make the body commence to move by twisting about the same screw. 

The method by which we have been conducted to the Principal Screws 
of Inertia shows how there are in general two, and only two, of these screws 
on the cylindroid. The homographic axis is the Pascal line, for the 
Hexagon AA BB CC , and thus we have a dynamical significance for 
Pascal s theorem. 

139. Determination of the Homographic Axis. 

The two principal screws of inertia must be reciprocal, and must also be 
conjugate screws of inertia ( 84). The homographic axis must therefore 
comply with the conditions thus prescribed. We have already shown ( 58) 
the condition that two screws be reciprocal, and ( 135) the condition that 
two screws be conjugate screws of inertia, and, accordingly, we see 

1. That the homographic axis must pass through 0, the pole of the 
axis of pitch. 

2. That the homographic axis must pass through , the pole of the 
axis of inertia. 

The points and having been already determined we have accordingly, 
as the simplest construction for the homographic axis, the chord joining 
and . 

140. Construction for Instantaneous Screws. 

The points and afford a simple construction for the instantaneous 
screw, corresponding to a given impulsive screw. The construction depends 
upon the following theorem ( 81): 

If two conjugate screws of inertia be regarded as instantaneous screws, then 
the impulsive screw corresponding to either is reciprocal to the other. 

Let A be an impulsive screw (fig. 22); if we join AO we obtain H, the 
screw reciprocal to A ; and if we join HO we obtain A , the conjugate screw 



142] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 129 

of inertia to H. But, as A is the only screw reciprocal to H, it is necessary, 
by the theorem just given, that an impulsive wrench on A must make the 
body commence to move by twisting about A . 




Fig. 22. 

As and are fixed, it follows from a well-known theorem, that as 
otherwise proved in 137, A and A form two homographic systems. 

141. Twist Velocity acquired by an Impulse. 

We can obtain a geometrical expression for the twist velocity acquired 
about A by a unit impulsive wrench on A (Fig. 22). 

It appears, from 90 (see also 147), that the twist velocity acquired 
on cc by an impulsive wrench on 77, is proportional to 



2 



The numerator being the virtual coefficient is proportional to AO.A H 
( 68), and as u^ is proportional to A O .A H ( 134), we see that the 
required ratio varies as AO -f- A O which itself varies as 

HO 
HO 

hence we obtain the following theorem : 

The impulsive wrench on A, of intensity proportional to HO, generates a 
twist motion about A , with velocity proportional to HO . 

The geometrical representation of the effect of impulsive forces is thus 
completely determined both as regards the instantaneous screw, and the 
instantaneous twist velocity acquired. 

142. Another Construction for the Twist Velocity. 

A still more concise method of determining the instantaneous screw can 
be obtained if we discard the points and , and introduce a new fixed 
point, ft, also on the homographic axis. 

B. 9 



130 



THE THEORY OF SCREWS. 



[142- 



Let X, Y (Fig. 23) be the two principal screws of inertia. Let A be 
an impulsive screw, and A the corresponding instantaneous screw. Draw 





Fig. 23. 

through -A the line AH parallel to XY. Join HA , and produce it to 
meet the homographic axis at fl. Let a be the twist velocity generated by 
an impulsive wrench of unit intensity at X, and let /3 be the corresponding 
quantity for F. 

It may be easily shown that the triangle AA X is similar to YA Cl, and 
that the triangle AA Y is similar to XA l ; whence we obtain 

A YflA 



__ 
AX~~ OF AY 

The unit wrench on A can be decomposed into components on X and F of 

respective intensities 

AY AX 

XY XY 



AX 



These will generate twist velocities 

AT 

a ~XY 

Let <w be the resulting twist velocity on A , then the components on X and F 
must be equal to the quantities just written ; whence 

A Y AY 



XY 

A X 
XY 



= a. 



and we obtain 



or, 



XY 

ZF 

CIA 
"OF 5 



a : /3 :: OF: 



we thus see that O is a fixed point wherever A and A may be. 



144J PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 131 

It also follows that 



is constant ; whence we have the following theorem : 

Draw through the impulsive screw A a ray AH parallel to the homographic 
axis, then the ray from H to a fixed point fl on the homographic axis will 
cut the circle in the instantaneous screw A , and the acquired twist velocity will 
be inversely proportional to CIA . 

If the twist velocity to be acquired by A from a unit impulsive wrench 
on A be assigned, then CIA is determined : there will be two screws A , and 
two corresponding impulsive screws, either of which will solve the problem. 
The diameter through Cl indicates the two screws about which the body will 
acquire the greatest and the least velocities respectively with a given 
intensity for the impulsive wrench. 

143. Twist Velocities on the Principal Screws. 

The quantities a and ft, which are the twist velocities acquired by unit 
impulsive wrenches on the principal screws, can be expressed geometrically 
as follows (Fig. 22) : 

Let to be the twist velocity acquired on A by the wrench on A, then, by 

the last article, 

aAY = (oA Y, 

/3AX=a,A X; 

A Y AY 
whence : ft :: ^ : ^ . 

This ratio is the anharmonic ratio of the four points X, Y, A, A , that is, of 
X, Y, 0, ; whence, finally, 

O Y OY 

* : P O X OX 

144. Another Investigation of the Twist Velocity acquired by 
an Impulse. 

We have just seen that 

<*AY=a>A Y, 



whence aft AX . AY= tfA X . A Y. 

Let fall perpendiculars AP, A P , HQ on the homographic axis (Fig. 24). 
Then, by the properties of the circle,. 

AX. AY : A X. A Y :: AP : A P ; 
so that a/3 A P = a>-A P . 

92 



132 THE THEORY OF SCREWS. 

By similar triangles, 



[144, 




whence 



or, as before ( 141), 



Fig. 24. 

OA . O H O H* 



CO X 



OH 
OH 



It will be noticed that, for this investigation, H may have been chosen 
arbitrarily on the circle. We thus see that, besides the two points and , 
there will be a system of pairs of points of which any one may be employed for 
finding the instantaneous screw, and for determining the instantaneous twist 
velocity. 

If we choose any two points (Fig. 25), fl and H , so that 




Fig. 25. 



then A being given, Al determines H, and Hfl determines A , while the 
twist velocity is proportional to O T/ -=- IH. We can suppose fl at infinity, 



145] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 133 

and thus obtain the construction used in 142. A similar construction is 
obtained when fl is at infinity. 

The two points, A and A , will divide the arc cut off by XY in a constant 
anharmonic ratio, for the pencil H (XQQ Y) always preserves the same 
anharmonic ratio as H moves round the circle. 

145. A Special Case. 

If 77 be an impulsive screw, arid if a be the corresponding instantaneous 
screw, it will not usually happen that when a. is the impulsive screw ij is the 
corresponding instantaneous screw. If, however, in even a single case, it be 
true that the impulsive screw and the instantaneous screw are interchange 
able, then the relation will be universally true. 

Let fl and 1 (Fig. 26) be a pair of points belonging to the system 
described in 144. Then A being given, A is found. If A is similarly to 




Fig. 26. 

determine A, then the figure shows that fl must lie on the polar of fl , 
and, consequently, fl and fl are conjugate points with respect to the circle; 
or, what comes to the same thing, they divide XY harmonically. The same 
must be true of each pair of points H and fl , and therefore of and , and 
we have the following theorem : 

If the points and be harmonic conjugates of the points where the homo- 
graphic axis intersects the circle, then every pair of instantaneous and impulsive 
screws on the cylindroid are interchangeable. 

We might, perhaps, speak of this condition of the system as one of 
dynamical involution. In this remarkable case an impulsive wrench of unit 
intensity applied to one of the principal screws of inertia will generate a 
velocity equal and opposite to that which would have been produced if the 
wrench had been applied to the other principal screw. The construction 



134 THE THEORY OF SCREWS. [145, 

for the pairs of related screws becomes still more simplified by the theorem, 

that 

When the system is one of dynamical involution, the chord joining an 
impulsive screw with its instantaneous screw passes through the pole of 
the homograpldc axis. 

We may take the opportunity of remarking, that dynamical involution 
is not confined to the system of the second order. It may be extended to a 
rigid body with any number of degrees of freedom, or even to any system of 
rigid bodies. Whenever it happens that the relation of impulsive screw and 
instantaneous screw is interchangeable in one case, it is interchangeable in 
every case. 

For, let O l , ... 6 n be the co-ordinates of an instantaneous screw, then ( 97) 
the corresponding impulsive screw has for co-ordinates, 

MI n u n a 

v ly ... v n ; 

PI Pn 

and if this latter were regarded as an instantaneous screw, then its impulsive 
screw would be 



3 * 

Pn 



but as this is to be only 
we must have 



which shows that if the theorem be true for one pair, it is true for all. The 
conditions, of course, are, that any one of the following systems of equations 
be satisfied : 

1/2 ,.2 ,,. 2 

, "i _ i III _ i U n 

~ Pi ~ P 2 ~ ~ Pn 

146. Another Construction for the Twist Velocity acquired 
by an Impulse. 

Reverting to the general case, we find that the chord A A (Fig. 27) is cut 
by the homographic axis at T, so that the square of the acquired twist 
velocity is proportional to the ratio of TA to TA . 

For, with the construction in 142, draw HQ parallel to AT; then, 



AT 



146] 



PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 



135 



but we showed, in the article referred to, that A l varies inversely as the 
acquired twist velocity, whence the theorem is proved. 




Fig. 27. 

This is, in one respect, the simplest construction, for it only involves the 
chord A A and the homographic axis. 

The chord A A must envelop a conic having double contact with the 
circle (Fig. 28), for this is a general property of the chord uniting two corre 
sponding points, A and A , of two homographic systems. Let / be the 




Fig. 28. 

point of contact of the chord and conic (Fig. 28). Then A A is divided 
harmonically in 7 and T ; for, if ZFbe projected to infinity, the two conies 
become concentric circles, and the tangent to one meets it at the middle 
point of the chord in the other ; the ratio is therefore harmonic, and must 
be so in every projection ; whence, 

AI _^AT 

A I A T 

but the last varies as the square of the twist velocity acquired, and hence we 
see that 



136 THE THEORY OF SCREWS. [146- 

The chord joining any impulsive screw A to the corresponding instantaneous 
screw A envelops a conic, and the point of contact, I, divides the chord into 
segments, so that the ratio of A I to A I is proportional to the square of the 
twist velocity acquired about A by the unit impulsive wrench on A. 

147. Constrained Motion. 

We can now give another demonstration of the theorem in 90, which is 
thus stated : 

If a body, constrained to twist about the screw a, be acted upon by an 
impulsive wrench on the screw 77, then the twist velocity acquired varies as 



W 2 

The numerator in this expression is the virtual coefficient of the two 
screws, and the denominator is the function of 134, which is proportional 
to the kinetic energy of the body when twisting about a with the unit of 
twist velocity. 

Let a and t] be represented by A and I respectively (Fig. 29), and let A 




Fig. 29. 

be the impulsive screw which would correspond to A if the body had been 
free to twist about any screw whatever on the cylindroid defined by A and 
A . Let K be reciprocal to A . 

The impulsive wrench on / is decomposed into components on K and A. 
The former is neutralized by the constraints ; the latter has the intensity 

KI 
KA 

whence the twist velocity &>, acquired by A , is ( 141) proportional to 

KI HO 
KA HO 



148] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 137 

but, by geometry, 



whence, 63, 



ft) GC 



x 



OA 

KI.HO .OA 
HO.HA .OA 

A .HA 

3C ~i> j 

which is the required result. 

148. Energy acquired by an Impulse. 

The kinetic energy acquired by a given impulse, using the same notation 
as before, is ( 91) proportional to 



Let A be the impulsive screw, and A the screw about which the body 
is constrained to twist. Draw the chord AOH (Fig. 30), then, as A varies, 




Fig. 30. 

while A is fixed, the virtual coefficient of A and A varies as A H (63). 
The square of this is proportional to AT, the length of the perpendicular 
from A on the tangent at H. If PQ be the axis of inertia, the value of u* 
is proportional to the perpendicular A Q, and, accordingly, the kinetic energy 
acquired is proportional to 

AT 



138 THE THEORY OF SCREWS. [148- 

Any ray through P, the intersection of the axis of inertia with the tangent 
at H, cuts the circle in two points, A and A", either of which will receive the 
same kinetic energy from the given impulse. 

149. Euler s Theorem. 

If the body be permitted to select the screw about which it will 
commence to twist, then, as already mentioned, 94, Euler s theorem states 
that the body will commence to move with a greater kinetic energy than if 
it be restricted to some other screw. By drawing the tangent from P (not, 
however, shown in the figure) we obtain the point of contact B, where it is 
obvious that the ratio of the perpendiculars on PH and PQ is a maximum, 
and, consequently, the kinetic energy is greatest. It follows from Euler s 
theorem that B will be the instantaneous screw corresponding to A as the 
impulsive screw. The line BH is the .polar of P, and, consequently, BH 
must contain , the pole of the axis of inertia. We are thus again led 
to the construction ( 140) for the instantaneous screw 5; that is, draw 
AOH, and then HO B. 

150. To determine a Screw that will acquire a given Twist 
Velocity under a given Impulse. 

The impulsive screw being given, and the intensity of the impulsive 
wrench being one unit, the acquired twist velocity ( 147) will vary as 
(Fig. 30), 

AfH 
A Q 

If, therefore, the twist velocity be given, this ratio is given. A must then lie 
on a given ellipse, with H as the focus and the axis of inertia as the directrix. 
This ellipse will intersect the circle in four points, any one of which gives a 
screw which fulfils the condition proposed in the problem. 

The relation between the intensity of the impulsive wrench and the twist 
velocity generated can be also investigated as follows : 

Let P, Q, R, S be points on the circle (Fig. 31) corresponding to four im 
pulsive screws, and let P , Q , R , S be the four corresponding instantaneous 
screws deduced by the construction already given. Let p, q, r, s denote the 
intensities of the impulsive wrenches on P, Q, R, S, which will give the units 
of twist velocity on P , Q , R, S . Supposing that impulsive wrenches on 
P, Q, R neutralize, then the corresponding twist velocities generated on 
P , Q , R must neutralize also. In the former case, the intensities must be 
proportional to the sides of the triangle PQR ; in the latter, the twist 



150] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 139 

velocities must be proportional to the sides of the triangle P Q R . 
Introducing another quantity d, we have 

rP Q = dPQ, 
qP R = dPR, 




The three other groups of equations are similarly obtained 
rQ S = aQS, qP S = cPS, rP S = bPS, 



sR Q = aRQ, sQ P = cQP, sR P = bRP. 

Whence we easily deduce 

ap = bq = cr = ds = hpqrs, 

where h is a new quantity. We hence obtain from the first equation 

P Q = hPQpq. 

As this is absolutely independent of R and S, it follows that h must be inde 
pendent of the special points chosen, and that consequently for any two 
points on the circle P and Q, with their corresponding points P and Q , we 
must have 

P Q 

PI 



140 THE THEORY OF SCREWS. [150- 

In the limit we allow P and Q to coalesce, in which case, of course, P and 
Q coalesce, and p and q become coincident ; but obviously we have then 

PQ : ML :: PX : LX, 
P Q : ML :: P Y : LY- 

P Q P Y LX 
whence PQ = PX = LY> 

and as P Foc -L and PX oc , 



we have finally p oc - T v . 

Li 

The result is, of course, the same as that of 141. Being given the 
impulsive screw corresponding to P, we find P by drawing PXL and LYP ; 
and then to produce a unit twist velocity on P , the intensity of the impul 
sive wrench on P must be proportional to LX -r- LY. It is obvious that by 
a proper adjustment of the units of length, force and twist velocity, LX 
may be the intensity of the impulsive wrench, and LY the acquired twist 
velocity. 

151. Principal Screws of the Potential. 

Let us suppose that a body having two degrees of freedom is in a position 
of stable equilibrium under the influence of a conservative system of forces. 
If the body be displaced by a small twist, it will no longer be in a position of 
equilibrium, and a wrench has commenced to act upon it. This wrench can 
always, by suitable composition with the reactions of the constraints, be 
replaced by an equivalent wrench on a screw of the cylindroid (see 96). 

For every point H, corresponding to a displacement screw, we have a 
related point, H , corresponding to the screw about which the wrench is evoked. 
The relation is of the one-to-one type, and it will now be proved that the 
system of screws H is homographic with the corresponding system H . The 
proof is obtained in the same manner as that already given in 137, for 
impulsive and instantaneous screws. 

Let E be a displacement screw about which a twist of unit magnitude 
evokes a wrench of intensity e on E ; let f be the similar quantity for 
another pair of screws, F and F . 

A twist of unit amplitude about H may be decomposed into components, 

HF HE 
EF EF 

about E and F, respectively. 



152] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 141 

These will evoke wrenches on E and F of the intensities 



HF HE 



EF EF 

respectively. But this pair of wrenches are to compound into a wrench of 
intensity h on H , and consequently we have 

H F HF 



H E _ 

S EF 



HF H F , 
whence ~ : , :: J : e. 



If we take another pair of points, K and K , we have 

KF H F K F 



^ 
HE : KE :: H E K E 

whence (HKFE) = (H K F E ). 

Thus, the anharmonic ratio of any four points in one system is equal to that of 
their correspondents, and the two systems are homographic. 

The homographic axis intersects the circle in two points, which are the 
principal screws of the potential, i.e. a twist about either evokes a wrench 
on the same screw. Of course this homographic axis is distinct from that 
in 139. But this homographic axis, like the former one, passes through the 
pole of the axis of pitch because the principal screws of the potential are 
reciprocal. 

152. Work done by a Twist. 

Suppose that the body, when in equilibrium under the system of forces, 
receives a twist of small amplitude of about any screw a, a quantity of work 
is expended, which we shall denote by 

Fv a -a 2 . 

In this, F is a constant, whose dimensions are a mass divided by the square 
of a time, and v* is a linear magnitude specially appropriate to the screw a, 
and depending also upon the system of forces ( 102). We may compare 
and contrast the three quantities,^, u a , v a : each is a linear magnitude 
specially correlated to the screw a. The first and simplest, p a , is the pitch 
of the screw, and depends on the geometrical nature of the constraints ; u a 
involves also the mass of the body, and the distribution of the mass relatively 
to a ; v a , still more complicated, depends also on the system of forces. 



142 



THE THEORY OF SCREWS. 



[153- 



153. Law of Distribution of v a . 



As we follow the screw a around the circle, it becomes of interest to study 
the corresponding variations of the linear magnitude v a . We have already 
found a very concise representation of p a and u a by the axis of pitch and the 
axis of inertia, respectively. We shall now obtain a similar representation 
of v a by the aid of the axis of potential. 

It is shown ( 102) that v a 2 must be a quadratic function of the co 
ordinates; we may therefore apply to this function the same reasoning as 
we applied to u a 2 ( 134). We learn that v a 2 is at each point proportional to 
the perpendicular on a ray, which is the axis of potential. 

Thus, if A (Fig. 32) be the screw, the value of v a 2 is proportional to AP, 
the perpendicular on PT\ if 0" be the pole of the axis of potential, then, 
as in 59, we can also represent the value of v a ~ by the product AO". A A . 




154. Conjugate Screws of Potential. 

In general the energy expended by a small twist from a position of 
equilibrium can be represented by a quadratic function of the co-ordinates 
of the screw. If, moreover, the two screws of reference form what are 
called conjugate screws of potential ( 100), then the energy is simply the 
sum of two square terms. The necessary and sufficient condition that the 
two screws shall be so related is, that their chord shall pass through 0". 

Another property of two conjugate screws of potential is also analogous 
to that of two conjugate screws of inertia. If A and A be two conjugate 
screws of potential, then the wrench evoked by a twist round A is reciprocal 
to A , and the wrench evoked by a twist around A is reciprocal to A. 



156] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 143 

155. Determination of the Wrench evoked by a Twist. 

The theorem just enunciated provides a simple means of discovering the 
wrench which would be evoked by a small twist which removes the body 
from a position of equilibrium. 

Let A (Fig. 33) be the given screw; join AO", and find H; then the 
required screw A must be reciprocal to H, and is, accordingly, found by 
drawing the chord HA through 0. 




Fig. 33. 

The axis 00" is of course the homographic axis of 151. We need not 
here repeat the demonstration of 141, which will apply, mutatis mutandis, 
to the present problem. We see that the ratio of the intensity of the 
wrench to the amplitude of the twist is proportional to 

HO 
HO" 

The other constructions of a like character can also be applied to this case. 

156. Harmonic Screws. 

If after displacement the rigid body be released, and small oscillations 
result, the present geometrical method permits us to study the resulting 
movements. 

It has been shown ( 130) that there are two special screws on the surface, 
each of which possesses the property of being a harmonic screw. If a body 
be displaced from rest by a small twist about a harmonic screw, and if it 
also receive any small initial twist velocity about the same screw, then the 
body will continue for ever to perform harmonic twist oscillations about the 
same screw. 

The two harmonic screws are X and T, where the circle is intersected 
by the axis passing through the pole of the axis of inertia , and the pole 
of the axis of potential 0" (Fig. 34). 



144 THE THEORY OF SCREWS. [156- 

For, suppose the body receives a small initial displacement about X, this 
will evoke a wrench on H, found by drawing XO"Y and YOH ( 155). But the 




Fig. 34. 

effect of a wrench on H will be to produce twist velocity about a screw found 
by drawing HOY and YO X, i.e. X itself ( 140). Hence the wrench evoked 
can only make the body still continue to twist about X, and harmonic 
vibration on X will be the result. Similar reasoning, of course, applies 
to Y. 

157. Small Oscillations in general. 

The initial displacement, and the initial twist velocity of the body, can 
always be decomposed into their respective components on X and Y. The 
resulting small oscillations can thus be produced by compounding simple 
harmonic twist oscillations about X and Y. 

If it should happen that and 0" become coincident, then every screw 
would be a harmonic screw. 

If and coincided, then every screw would be a principal screw of 
inertia ( 86). 

If and 0" coincided, then every screw would be a principal screw of 
potential. 

158. Conclusion. 

The object proposed in this Chapter has now been completed. It has 
been demonstrated that the representative circle affords a concise method 
of exhibiting many problems in the Dynamics of a Rigid System with two 
degrees of freedom, so long as the body remains near its initial position. 
The geometrical interest of the enquiry is found mainly to depend on the 
completely general nature of the constraints. If the constraints be specialized 
to those with which mechanical problems have made us familiar, it will 



158] PLANE REPRESENTATION OF DYNAMICAL PROBLEMS. 145 

frequently be found that the geometrical theory assumes some extreme and 
uninteresting type. For instance, a case often quoted as an illustration of 
two degrees of freedom, is that of a body free to rotate around an axis, 
and to slide along it. The representative circle has then an infinite 
radius, and the finite portion thereof is merely a ray perpendicular to the 
axis of pitch. The geometrical theory then retains merely a vestige of 
its interest. 



B. 10 



CHAPTER XIII. 

THE GEOMETRY OF THE CYLINDROID. 

159. Another investigation of the Cylindroid. 

The laws of the composition of twists and wrenches are of such funda 
mental importance in the present subject that the following method* of 
investigating the cylindroid seems worthy of attention. This method is not, 
however, presented as a substitute for that already given (Chap. II.) which is 
certainly both more simple and more direct. 

Let a and /3 be any two screws, then if a body receives a twist about a, 
followed by another twist about /3, the position arrived at could have been 
reached by a single twist about a third screw 7. If the amplitudes of the 
twists about a and ft are given, then the position of 7, as well as the ampli 
tude of the resultant twist thereon, are, of course, both determined. If, 
however, the amplitudes of the twists on a and ft are made to vary while 
the screws a and ft themselves remain fixed, then the position of 7, no less 
than the amplitude of the resultant twist, must both vary. It has however 
been shown in 9 that the position and pitch of 7 remain constant so long as 
the ratio of the amplitudes of the twists about a and ft remains unchanged. 
As this ratio varies, the position of 7 will vary, so that this position is a 
function of a single parameter ; and, accordingly, 7 must be restricted to be 
one of the generators of a certain ruled surface S, which includes a and ft as 
extreme cases in which the ratio is zero and infinity respectively. 

Let 6 l be a screw which is reciprocal both to a and ft, then 6 l must also 
be reciprocal to every screw 7 on S. Let 2 , 3> # 4 be three other screws also 
reciprocal to S. Since a screw is defined by five conditions, it is plain that a 
screw which fulfils the four conditions of being reciprocal to lf # 2 > 0s> $4 will 
have one disposable parameter, and must, therefore, be generally confined to 
a certain ruled surface. This surface must include S, inasmuch as all the 
screws on S are reciprocal to l} 2 , 3 , 4 ; but further, it cannot include any 

* Proceedings of the Royal Irish Academy, 2nd Ser., Vol. iv. p. 518 (1885). 



159] THE GEOMETRY OF THE CYLINDROID. 147 

screw e not on S ; for as e and any screw 7 on S are reciprocal to lt # a , 3 ,0 4 , 
it will follow that any screw on the surface made from e and 7, just as S is 
made from a and ft, must also be reciprocal to 6 l) 2 , 3 , 4 . As 7 may be 
selected arbitrarily on S, we should thus find that the screws reciprocal to 
&i, &2, $s> #4 were not limited to one surface, but constituted a whole group of 
surfaces, which is contrary to what has been already shown. It is therefore 
the same thing to say that a screw lies on S, as to say that it is reciprocal to 
0i, 6* d. ( 24). 

Since the condition of reciprocity involves the pitches of the two screws 
in an expression containing only their sum, it follows that if all the pitches 
on 1} 2 , 3 , # 4 be diminished by any constant in, and all those on S be 
increased by m, the reciprocity will be undisturbed. Hence, if the pitches 
of all the screws on S be increased by + m, the surface so modified will still 
retain the property, that twists about any three screws will neutralize each 
other if the amplitudes be properly chosen. 

We can now show that there cannot be more than two screws of equal 
pitch on 8 ; for suppose there were three screws of pitch m, apply the 
constant m to all, thus producing on S three screws of zero pitch. It must 
therefore follow that three forces on 8 can be made to neutralize ; but this is 
obviously impossible, unless these forces intersect in a point and lie on a 
plane. In this case the whole surface degrades to a plane, and the case is a 
special one devoid of interest for our present purpose. It will, however, be 
seen that in general S does possess two screws of any given pitch. We can 
easily show that a wrench can always be decomposed into two forces in such 
a way that the line of action of one of these forces is arbitrary. Suppose 
that 8 only possessed one screw A, of pitch m. Reduce this pitch to zero ; 
then any other wrench must be capable of decomposition into a force on X 
(i.e. a wrench of pitch zero), and a force on some other line which must lie 
on S; therefore in its transformed character there must be a second screw 
of zero pitch on S, and, therefore, in its original form there must have been 
two screws of the given pitch m. 

Intersecting screws are reciprocal if they are rectangular, or if their 
pitches be equal and opposite ; hence it follows that a screw 6 reciprocal to 
S must intersect 8 in certain points, the screws through which are either at 
right angles to or have an equal and opposite pitch thereto. 

From this we can readily show that S must be of a higher degree than 
the second ; for suppose it were a hyperboloid and that the screws lay on 
the generators of one species A, a screw which intersected two screws 
of equal pitch m must, when it receives the pitch m, be reciprocal to the 
entire system A. We can take for one of the generators on the hyper 
boloid belonging to the species B ; will then intersect every screw of the 

10-2 



148 THE THEORY OF SCREWS. [159 

surface ; it must also be reciprocal to all these ; and, as there are only two 
screws of the given pitch, it will follow that 6 must cut at right angles every 
generator of the species A. The same would have to be true for any other 
reciprocal screw < similarly chosen ; but it is obvious that two lines 6 and < 
cannot be found which will cut all the generators at right angles, unless, 
indeed, in the extreme case when all these are coplanar and parallel. In the 
general case it would require two common perpendiculars to two rays, which 
is, of course, impossible. We hence see that S cannot be a surface of the 
second degree. 

We have thus demonstrated that 8 must be at least of the third degree 
in other words, that a line which pierces the surface in two points will pierce 
it in at least one more. Let a and ft be two screws on 8 of equal pitch in, 
and let 6 be a screw of pitch m which intersects a and ft. It follows that 

6 is reciprocal both to a and ft, and therefore it must be reciprocal to 
every screw of 8. Let 6 cut 8 in a third point through which the screw 7 is 
to be drawn, then 6 and 7 are reciprocal ; but they cannot have equal and 
opposite pitches, because then the pitch of 7 should be equal to that of a 
and ft. We should thus have three screws on the surface of the same pitch, 
which is impossible. It is therefore necessary that 6 shall always intersect 7 
at right angles. From this it will be easily seen that 8 must be of the 
third degree ; for suppose that 6 intersected 8 in a fourth point, through 
which a screw 8 passed, then 6 would have to be reciprocal to 8, because it is 
reciprocal to all the screws of $; and it would thus be necessary for 9 to be 
at right angles to 8. Take then the four rays a, ft, 7, 8, and draw across 
them the two common transversals 6 and <. We can show, in like manner, 
that (j) is at right angles to 7 and 8. We should thus have 6 and </> as two 
common perpendiculars to the two rays 7 and 8. This is impossible, unless 

7 and 8 were in the same plane, and were parallel. If, however, 7 and 8 be 
so circumstanced, then twists about them can only produce a resultant twist 
also parallel to 7 and 8, and in the same plane. The entire surface 8 would 
thus degenerate into a plane. 

We are thus conducted to the result that 8 must be a ruled surface 
of the third degree, and we can ascertain its complete character. Since any 
transversal 6 across a, ft, and 7 must be a reciprocal screw, if its pitch be 
equal and opposite to those of a and ft, it will follow that each such trans 
versal must be at right angles to 7. This will restrict the situation of 7, 
for unless it be specially placed with respect to a and ft, the transversal 6 
will not always fulfil this condition. Imagine a plane perpendicular to 7, 
then this plane contains a line / at infinity, and the ray 6 must intersect / as 
the necessary condition that it cuts 7 at right angles. As 6 changes its 
position, it traces out a quadric surface, and as I is one of the generators of 
that quadric, it must be a hyperbolic paraboloid. The three rays a, ft, 7, 



159] THE GEOMETRY OF THE CYLINDROID. 149 

belonging to the other system on the paraboloid must also be parallel to a 
plane, being that denned by the other generator / , in which the plane at 
infinity cuts the quadric. 

Let PQ be a common perpendicular to a and 7, then since it intersects 
7 at right angles, it must also intersect / ; and since PQ cuts the three 
generators of the paraboloid a, 7 and /, it must be itself a generator, and 
therefore intersects ft. But a, ft, 7 are all parallel to the same plane, and 
hence the common perpendicular to a and 7 must be also perpendicular to ft. 
We hence deduce the important result, that all the screws on the surface S 
must intersect the common perpendicular to a and ft, and be at right angles 
thereto. 

The geometrical construction of S is then as follows: Draw two rays a and 
ft, and also their common perpendicular X. Draw any third ray 6, subject 
only to the condition that it shall intersect both a and ft. Then the common 
perpendicular p to both and X will be one of the required generators, 
and as 6 varies this perpendicular will trace out the surface. It might 
at first appear that there should be a doubly infinite series of common 
perpendiculars p to X and to 6. Were this so, of course S would not be 
a surface. The difficulty is removed by the consideration that every trans 
versal across p, a, ft is perpendicular to p. Each p thus corresponds to a 
singly infinite number of screws 9, and all the rays p form only a singly 
infinite series, i.e. a surface. 

A simple geometrical relation can now be proved. Let the perpendicular 
distance between p and a be d lt and the angle between p and a be A l ; let d z 
and A 2 be the similar quantities for p and ft, then it will be obvious that 

rfj : d z : : tan A l : tan A 2 ; 
or rfj + d 2 : d 1 d. 2 : : sin (A^ + A 2 ) : sin (A l A 2 ), 

if z be the distance of p from the central point of the perpendicular h 
between a and ft ; and if e be the angle between a and ft, and 6 be the 
angle made by p with a parallel to the bisector of the angle e, then we have 
from the above 

z : h : : sin 2< : sin 2e. 

The equation of the surface 8 is now deduced for 

oc 
tan 6 = - ; 

y 

whence we obtain the equation of the cylindroid in the well-known form 

z (# 2 + */*) = xy. 

sm 2e * 

The law of the distribution of pitch upon the cylindroid can also be deduced 



150 THE THEORY OF SCREWS. [159 

from the same principles. If a and yS are screws of zero pitch, then any 
reciprocal transversal 6 will be also of zero pitch ; and as p must be reciprocal 
to 6, it will follow that the pitch of p must be equal to the product of the 
shortest perpendicular distance between p and 6, and the tangent of the 
angle between the two lines. In short, the pitch of p must simply be equal 
to what is sometimes called the moment between p and 6. 

We are also led to the following construction for the cylindroid. 

Draw a plane pencil of rays and another ray L, situated anyivhere. Then 
the common perpendiculars to L and the several rays of the pencil trace out 
the cylindroid. 

I have already mentioned (p. 20) that the first model of the cylindroid 
was made by Plucker in illustration of his Neue Geometric des Raumes. The 
model of the surface which is represented in the Frontispiece was made from 
my design by Sir Howard Grubb, the cost being defrayed by a grant from 
the Scientific Fund of the Royal Irish Academy. A hollow cylinder was 
mounted on a dividing engine and holes were drilled at the calculated points. 
Silver wires were then stretched across in the positions of the generators 
and a beautiful model is the result. 

The equation to the tangent cone drawn from the point x, y , z to 
the surface, 

z (x 2 4- y 2 ) - 2mxy = 0, 



is of the fourth degree and is given by equating to zero the discriminant 
of the following function in &>, 

to 3 (xz- zx) - <u 2 [yz -zy + 2m (x - x }} + o> {xz - zx + 2m (y - y )} + zy - yz. 

This cone has three cuspidal edges, and accordingly the model exhibits 
in every aspect a remarkable tricuspid arrangement. 

I here give the details of the construction of the much simpler model of 
the cylindroid figured in Plate II.* A boxwood cylinder, O m lo long and 
O m> 05 in diameter, is chucked to the mandril of a lathe furnished with a 
dividing plate. A drill is mounted on the slide rest, and driven by overhead 
gear. The parameter p a -p ft (in the present case O ^OGG) 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 

* See Transactions of Royal Irish Academy, Vol. xxv. p. 216 (1871) ; and also Phil. Mag. Vol. 
XLII. p. 181 (1871); also B. A. Report, Edinburgh, 1871. 




THE CYLINDROID. 



To face IL 150 



159] 



THE GEOMETRY OF THE CYLINDROID. 



151 



axis of the cylinder at right angles. The following table will enable the 
work to be executed with facility. I is the angle of 13 : 



z 


I 


90-1 


180 + 1 


270 -I 


o-o 





90 


180 


270 


17-4 


5 


85 


185 


265 


34-2 


10 


80 


190 


260 


50-0 


15 


75 


195 


255 


64-3 


20 


70 


200 


250 


76-5 


25 


65 


205 


245 


86-6 


30 


60 


210 


240 


94-0 


35 


55 


215 


235 


98-5 


40 


50 


220 


230 


100-0 


45 


45 


225 


225 



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 O m- 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 n 066 cos 21 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. 

Among the various other representations of the cylindroid I can now do 
no more than refer to an ingenious plan described by Goebel in his Neueren 
Statik, by which a model of this surface in card-board can be made with 
facility. There is also a model in the collection of the Cavendish Laboratory 
at Cambridge, and another belonging to the Mathematical Society of London, 
which, like that figured in Plate II., was made by myself. Sir Howard Grubb 
has also made a second model with the same dimensions as that figured 
in the frontispiece but mounted in a different manner. This exquisite 
exhibition of a ruled surface is the property of Mr G. L. Cathcart, Fellow of 
Trinity College, Dublin. 

A suggestive construction for the cylindroid has been also given by 
Professor G. Minchin in his well-known book on Statics, and we have already 
mentioned (note to 50) the construction given by Mr Lewis. 



152 



THE THEORY OF SCREWS. 



[160 



160. Equation to plane section of Cylindroid. 

Each generator of the cylindroid is the abode of a certain screw, and 
accordingly each point in a plane section will lie on one screw, and generally 
on only one. We may, accordingly, regard the several points of the cubic as 
in correspondence with the several screws on the cylindroid. It will often be 
convenient to speak of the points on the section as synonymous with the 
screws themselves which pass through those points. 

We must first investigate the equation* to the cubic curve produced by 
cutting the cylindroid by a plane situated in any arbitrary position. 




Fig. 35. 

Let OX and OF (Fig. 35) be the two principal screws of the cylindroid of 
which OH is the nodal line. Let XYl be the arbitrary plane of section. 
The position of this plane is defined by the magnitudes h, a, (3, whereof h 
is the length of the perpendicular from on XY, a. is the angle between OR 
and OX, and ft is the angle ORl, or the inclination of the plane of section 
to the principal plane of the cylindroid. 

Draw through H the line 1$ parallel to XY; then we shall adopt IN 
as the new axis of x and SIR as the new axis of y, so that if P be any point 
on the surface, we have PN = y and IN = x. The dotted letters, x , y , z 
refer to the original axes of the cylindroid. Let fall PT perpendicular on 
the plane of OXY, and TM perpendicular to XY. Then we have MN= 

whence 

y + z cosec ft = h sec /3 (i), 

* Transactions of the Eoyal Irish Academy, Vol. xxix. p. 1 (1887). 



160] THE GEOMETRY OF THE CYLINDRO1D. 153 

while, if 6 be the angle XOT, and OT be denoted by r, we have 

x = r sin(0 - a), 

or x = y cos a x sin a (ii) ; 

but, obviously, 

OR = r cos (0 - a) + M T ; 

whence h = x cos a 4- y sin a + z cot /3 (iii). 

Solving the equations (i), (ii), (iii), we obtain 

x e = x sin a + y cos /3 cos a, 
y = + x cos a + y cos /3 sin a, 
z = h tan /3 y sin /3. 
It appears from these that 

# 2 + y z = x* + ifcos 2 ft, 

a; ?/ = #?/ cos /3 cos 2 a + (y 2 cos 2 /3 # 2 ) sin a cos a. 
The equation of the cylindroid gives 

z (x z + y 2 ) = Zmx y ; 

whence we deduce, as the required equation of the section, 
(h tan ft - y sin /3) (# 2 + y z cos 2 /3) 

= 2m3ry cos ft cos 2a + 2m sin a cos a (y 2 cos 2 ft a?); 
or, arranging the terms, 

sin /3 cos 3 fty 3 + sin yS?/a; 2 (m sin 2a + h tan /3) # 2 + Zmxy cos /3 cos 2 

+ (m sin 2a cos 2 ft h sin /3 cos /3) y J = 0. 

It is often convenient to use the expressions 

x = h tan (6 0) m sin 20 cot ft tan (0 a), 
y = hsecft msin 20cosec/3, 

from which, if be eliminated, the same equation for the cubic is obtained ; 
or, still more concisely, we may write 

x = y cos ft tan (0 a), 
y = h sec ft m sin 20 cosec ft. 
This cubic has one real asymptote, the equation of which is 

ysinft = m sin 2a + h tan ft, 
and the asymptote cuts the curve in the finite point for which 

x = tan 2a (h + m sin 2a cot ft). 
The value of at this point is a. 



154 



[160, 




Fig. 36. 

EXPLANATION OF FIG. 36. 
General Section of the Cylindroid, showing 

(1) Cubic with the double point 0. 

(2) Asymptote of the cubic. 

(3) The parabola, which is the envelope of the chords joining screws of equal pitch. 

(4) Hyperbola having triple contact with the cubic, being envelope of reciprocal chords. 

(5) Section of the principal plane. It is a tangent to the hyperbola. 

(6) A tangent to the parabola, showing two screws, P and Q, of equal pitch. 

(7) Common tangents to the parabola and the cubic, touching the latter at the two 
principal screws. 

(8) Any tangent to the hyperbola intersects the cubic in three points, two of which belong 
to reciprocal screws (not shown). 

Equations of Cubic. Equation of Parabola. Equations of Hyperbola. 

a; = -9f/tan(0-25 ), /_ x \ 2 a; = 1-6 21-6 sec047 5 tan 0, 

y = 20 -66 sin 20. V ~ \ + 15J y = - 12-8 + 32 8 sec <f>. 



161] THE GEOMETRY OF THE CYLINDROID. 155 

In Fig. 36 will be found a drawing of this curve. The following are the 
values of the constants adopted : 

a = 25; /3 = 26; h = lS; wi = 28 9; 
with which the equations become 

= % tan (0-25), 
y = 20 - 66 sin 20. 

The curve was plotted down on " papier millimetrique," and has been copied 
in reduced size in the figure. The constants were selected after several 
trials, in order to give a curve that should be at once characteristic, and 
of manageable dimensions. 

The distribution of pitch upon the screws of the cylindroid is of 
fundamental importance in the theory, so that we must express the pitches 
appropriate to the several points on the cubic. 

Let p denote the pitch ; then, from the known property of the cylindroid, 

p=p + mcos 26, 

where p Q is a constant. Transforming this result into the co-ordinates of 
the point on the cubic, we have 

(x 2 if cos 2 ft) cos 2a + 2xy cos 8 sin 2a 

/\1 /i /vvi v_ * 

P ~ P ^ + 2/ 2 cos 2 /3 

161. Chord joining Two Screws of Equal Pitch. 

As the pitches of the two screws, defined by + and 6, are equal, the 
chord in question is found by drawing the line through the points x , y and 
x", y", respectively, where 

x = y cos ft tan (9 a), 
y = h sec ft m cosec ft sin 20, 
x" = y" cos ft tan ( a), 
y" = h sec ft + m cosec ft sin 20. 

After a few reductions, the required equation is found to be 
xm (cos 20 + cos 2a) + y (h sin ft + m cos ft sin 2a) 

- h- tan ft + m 2 cot ft sin 2 20 - 0. 
If this chord passes through the origin, then 

- h* tan- ft + ??i 2 sin 2 20 = ; 
or, h tan m sin 20 = 0. 

But this is obviously necessary ; for from the geometry of the cylindroid it is 
plain that must then fulfil the required condition. 



156 THE THEORY OF SCREWS. [161, 

We can also determine the chord in a somewhat different manner, which 
has the advantage of giving certain other expressions that may be of 
service. 

Let U = be the cubic curve. 

Let V= be the equation of the two straight lines from the origin to the 
points of intersection with the two equal pitch screws + 6. 

Let L = be the chord joining the two intersections of U and V, distinct 
from the origin : this is, of course, the chord now sought for. Then we must 
have an identity of the type 

cU=VX 



where c is some constant. For the conditions L = and V= imply U=0, and 
L cuts U in three points, two of which lie on V, and the third point, called /, 
must lie on X. The line X is otherwise arbitrary, and we may, for con 
venience, take it to be the line 11 from the origin to /. The product VX 
thus contains only terms of the third degree, and accordingly the terms of 
the second degree in U must be sought in LY. 

Let U=u 3 +u 2 where u 3 and u 2 are of the third and second degrees 
respectively, then cu. 2 must be the quadratic part of the product L Y. As L 
does not pass through the origin, it must have an absolute term, conse 
quently Y must not contain either an absolute term or a term of the first 
degree. If, therefore, c be the absolute term in L, it is plain that Y must 
be simply u 2 , and we have accordingly, 

c (u s + MS) = VX + (L + c) u 2 , 

where L denotes the value of L without the absolute term : we have con 
sequently the identity 

cu 3 = VX + L u 2 . 

In this equation we know u z , u 3 , V, and the other quantities have to be found. 

If we substitute 

x = y cos /3 tan (a + 6), 

we make V vanish, and representing L by \x + py, we find 

X cos ft tan (a 6) p = c T~ 5 - ^ > 

h tan /9 + ra sin 20 

and after a few steps 

\ m cos 2a + tn cos 20 



c - h* tan /3 + ?/i 2 cot /3 sin 2 2 

p _ h sin /3 + m cos ft sin 2a 
c ~~ h 2 tan /3 + ??i 2 cot ft sin 2 2 



162J THE GEOMETRY OF THE CYLINDROID. 157 

We can also obtain X ; for let it be Px+ Qy, then severally identifying the 
coefficients of a? and y 3 , we have, 

P = m sin 2a 4- h tan ft, 
Q = m cos ft (cos 2a + cos 20) ; 
finally, resuming the various results, we obtain the identity 



wherein, 

c = - h 2 tan ft + m 2 cot /3 sin 2 20, 
U = sin/3 cos 2 /3?/ 3 + sin Pyx?, 
(m sin 2a + A tan /3) a? 2 + 2mxy cos /3 cos 2a 
+ (m sin 2a cos 2 ft h sin ft cos /3) t/ 2 . 

F= wz# 2 (cos 2a + cos 2$) + 2//m/ sin 2a cos ft + my 2 cos 2 ft (cos 20 cos 2a), 
X = oc(m sin 2a + A tan /3) my cos ft (cos 2a + cos 20), 
i = TO# (cos 2a + cos 20) + y (h sin /3 + m cos /3 sin 2a) 

- A 2 tan /3 + m 2 cot sin 2 20, 
F= a? ( m sin 2a A tan /3) -f 2?n#y cos ^ cos 2a 

+ 2/ 2 (m sin 2a cos 2 /3 h sin y3 cos y3). 

162. Parabola. 

The screws reciprocal to a cylindroid intersect two screws of equal pitch 
on the surface. Any chord in the section which cuts the cubic in two points 
of equal pitch must thus be the residence of a screw reciprocal to the surface; 
accordingly the chord 

mx (cos 2a + cos 20) + y (h sin ft + m cos ft sin 2a) 

- h- tan ft + m? cot ft sin 2 20 = 0, 
when it receives a pitch equal to 

p Q m cos 20, 
forms a screw reciprocal to the cylindroid. 

It is easily shown that the envelope of this chord is a parabola; differ 
entiating with respect to we have 

x = 2m cot ft cos 20. 
Eliminating we obtain 

x z + 4<mx cot ft cos 2<z + 4y (h cos ft + m cos ft cot ft sin 2a) - 4h? + 4m 2 cot 2 ft = 0. 
The vertex of the parabola is at the point 

x = - 2m cot ft cos 2a ; y = h sec ft m cosec ft sin 2a. 



158 THE THEORY OF SCREWS. [162 

The latus rectum is 

4 cos /3 (h + m cot /3 sin 2a). 

The values of the two equal pitches (p) on the pair of screws are thus 
expressed in terms of the abscissa x of the point in which the chord touches 
the envelope by means of the equation 



From any point P, on the cubic, two tangents can be drawn to the 
parabola. Each of these tangents must intersect the cubic in a pair of screws 
of equal pitch. One tangent will contain the other screw whose pitch is equal 
to that of P. The second tangent passes through two screws of equal pitch 
in the two other points which, with P, make up the three intersections with 
the cubic. As the principal screws of the cylindroid are those of maximum 
and minimum pitch respectively, it follows that the tangents at these points 
will also touch the parabola. These common tangents are shown in the 
figure. 

This parabola is drawn to scale in Fig. 36. The equation employed was 

2 



5 + 

When the figure was complete, it was obvious that the parabola touched 
the cubic, and thus the following theorem was suggested : 

The parabola, which is the envelope of chords joining screws of equal pitch, 
touches the cubic in three points. 

The demonstration is as follows : To seek the intersections of the 
parabola with the cubic, we substitute, in the equation of the parabola, the 
values 

x = h tan (6 a) m cot /3 sin 20 tan (9 a), 

y = h sec /3 m cosec /3 sin 20, 

This would, in general, give an equation of the sixth degree for tan 9. It 
will, however, be found in this case that the expression reduces to a perfect 
square. The six points in which the parabola meets the cubic must thus 
coalesce into three, of which two are imaginary. The values of 9 for these 
three points are given by the equation 

h tan (9 - a) - m cot (sin 26 tan (9 - a) + 2 cos 20} = 0. 

We can also prove geometrically that the parabola touches the cubic at 
three points. 

In general, a cone of screws reciprocal to the cylindroid can be drawn 
from any external point. If the point happen to lie on the cylindroid, 



162] THE GEOMETRY OF THE CYLINDROID. 159 

the cone breaks up into two planes. The nature of these planes is easily 
seen. One of them, A, must be the plane perpendicular to the generator 
through ; the other, B, is the plane containing 0, and the screw of equal 
pitch to that of the screw through 0. These planes intersect in a ray, L, 
and it must first be shown that L is a tangent to the cyliridroid. 

Any ray intersecting one screw on a cylindroid at right angles must cut 
the surface again in two screws of equal pitch ; consequently L can only 
meet the surface in two distinct points, each of which has the pitch of the 
generator through 0. It follows that L must intersect the surface at two 
coincident points 0, i.e. that it is a tangent to the cylindroid at 0. 

Let any plane of section be drawn through 0. This plane will, in 
general, intersect A and B in two distinct rays : these are the two screws 
reciprocal to the cylindroid, and they are accordingly the two tangents from 
to the parabola we have been discussing. The only case in which these 
two rays could coalesce would occur when the plane of section was drawn 
through L ; but the two tangents to a parabola from a point only coalesce 
when that point lies on the parabola. At a point where the parabola meets 
the cubic, L must needs be a tangent both to the parabola and to the cubic, 
which can only be the case if the two curves are touching. We have thus 
proved that the parabola must have triple contact with the cubic. 

There are thus three points on the cubic which have the property that 
the tangent intersects the curve again in a point of equal pitch to that of the 
point of contact. We thus learn that all the screws of a four-system which 
lie in a plane touch a parabola having triple contact with the reciprocal 
cylindroid. 

From any point P, on the cubic, two tangents can be drawn to the 
parabola. Each of these tangents must intersect the cubic in a pair of screws 
of equal pitch. One tangent will contain the two screws whose pitch is equal 
to that of P. The other tangent passes through two screws of equal pitch 
in the two other points, which, with P, make up the three intersections 
with the cubic. 

As the principal screws of the cylindroid are those of maximum and 
minimum pitch, respectively, it follows that the tangents at these points 
will also touch the parabola. These common tangents are shown in Fig. 36. 

From the equation of the cylindroid, 

z (a? + 2/ 2 ) = 2ma?y, 

it follows that the plane at infinity cuts the surface in three straight lines 
on the planes, 

* = 0, 

x iy = 0. 



160 THE THEORY OF SCREWS. [162- 

The line at infinity 011 the plane of z is of course intersected by all the real 
generators of the cylindroid, inasmuch as they are parallel to z. The points 
at infinity on the planes x iy = are each the residence of an imaginary 
screw, also belonging to the surface. The pitches of both these screws are 
infinite. 

We may deduce the two screws of infinite pitch on the surface in another 
way. The equations of a screw are 

y = x tan 6, 
z = m sin 20, 

while the pitch is 

p + m cos 20. 

If tan 6 be either + i, we find both z infinite and the pitch infinite. We 
thus see that through the infinitely distant point /, on the nodal line of the 
cylindroid, two screws belonging to the surface can be drawn, just as at any 
finite point. The peculiarity of the two screws through / is, that their 
pitches are equal, i.e. both infinite, and this is not the case with any other 
pair of intersecting screws. 

It is now obvious why the envelope just considered turned out to be 
a parabola rather than any other conic section. Every plane section will 
have the line at infinity for a transversal cutting two screws of equal pitch ; 
the envelope of such transversals must thus have the line at infinity for 
a tangent, i.e. must be a parabola. 

163. Chord joining Two Points. 

If 6 and 6" be the angles by which two points on the cubic are defined, 
then the equation to the chord joining those points is 



where A = 2m cos (0 - a) cos (9" - a) cos (0 + 0"), 

B = hsm/3+m cos j3 cos (6 r + 0") sin (2a - & - & } 

- m cos sin (0 + 0") cos (ff - 0"), 
C = - tan /3 (h - m cot /3 sin 20 ) (h-m cot j3 sin 2(9"). 

If in these expressions we make + 6" = 0, we obtain the equation for the 
chord joining screws of equal pitch, as already obtained. 

We shall find that, in particular sections, these expressions become con 
siderably simplified. Suppose, for example, that the plane of section be a 
tangent plane to the cylindroid. The cubic then degenerates to a straight 
line and a conic. The condition for this will be obvious from the equation 



164] THE GEOMETRY OF THE CYLINDROID. 161 

of the cubic. If the coefficient of a? become zero, the required decomposition 
takes place, for y is then a factor. The necessary and sufficient condition 
for the plane of section being a tangent, is therefore 

ra sin 2a + h tan /3 = 0. 

When this is the case, the three expressions of A, B, C may be divided by 
a common factor, 

2m cos (6 a) cos (6" a). 
and we have 

A = cos (& + 0"), 

B = - cos /3 sin (& + 0"), 

G = - 2m cot 13 sin (a + ) sin (a + 0"). 

If the screws be of equal pitch, 9 + 0" = 0, the coefficient of y disappears, 
and we see that all the chords are merely lines parallel to the axis of y, which 
is parallel to one of the axes of the ellipse. 

The equation to the chord then becomes 

(cos 2a + cos 20) [x + m cot ft (cos 2a cos 20)} = 0. 

For a given value of x there are two values of corresponding to the two 
chords that can be drawn through the point. One of these chords is parallel 
to y, and has a obtained from the equation 

cc + m cot /3 (cos 2a cos 20) = 0. 

7T 

The other value of is -- a, from the equation 

2 

cos 2a+ cos 20 = 0. 

This is independent of x, as might have been foreseen from the fact that 
the two screws of equal pitch are in this case the line in the section and the 
other screw of equal pitch. The latter cuts the section in a certain point, and, 
of course, all chords through this point meet the curve in two screws of equal 
pitch. 

164. Reciprocal Screws. 

Another branch of the subject must now be considered. We shall first 
investigate the following general problem : 

From any point, P, a series of transversals is drawn across each pair 
of reciprocal screws on the cyliiidroid. It is required to determine the cone 
which is the locus of these transversals. We shall show that this is a cone 
of the second degree. 

B. 11 



162 THE THEORY OF SCREWS. [164, 

Let a, /3, 7 be the co-ordinates of the point. Then the plane through 
this point, and the generator of the cylindroid defined by the equations 

y = x tan 0, 
z = in sin 20, 
is (y x tan 6) (7 m sin 20) = (/3 a tan 6) (z m sin 20) ; 

or, if we arrange in powers of tan 0, we obtain 

A tan 3 + B tan 2 + Otan + D = 0, 

in which 

A = VLZ yx ; D = <yy (3z, 

B = yy /3z + 2mx 2ma ; C = a.z yx + 2m/3 2my. 

If the same transversal also crosses the generator defined by , then, 

A tan 3 + B tan 2 & + G tan & + D = 0. 
When the two screws defined by and are reciprocal, 

tan tan = H, 
when H is a constant. 

By eliminating and rejecting the factor 

tan - tan & 

we obtain 

- A*H 3 + A CH* - BDH + D n - = 0. 

And as this is of the second degree in x, y, z, the required theorem has been 
proved. 

All these cones must pass through the centre of the cylindroid, inasmuch 
as the two principal screws of the cylindroid are reciprocal. If a constant 
be added to the pitches of all the screws on the cylindroid, then the pairs 
of reciprocals alter, inasmuch as H alters. The cone changes accordingly, 
and thus there would be through each point a family of cones, all of which, 
however, agree in having, as a generator, the ray from the vertex to the 
centre of the cylindroid. Thus, even when the cylindroid is given, we must 
further have the pitch of a stated screw given before the cone becomes 
definite. This state of things may be contrasted with that presented by the 
cone of reciprocal screws which may be drawn through a point. The latter 
depends only upon the cylindroid itself, and is not altered if all the pitches 
be modified by a constant increment. 



165] THE GEOMETRY OF THE CYLINDROID. 163 

The discriminant of the cubic 

- A n -H 3 + A CH* - SDH + D* = 
is 

A -D* (A 2 D* + B 3 D + -fa AC 3 - &C* - f ABCD). 

Omitting the factor A~D 2 , we have, for the envelope of the system of cones, 
the cone of the fourth order, found by equating the expression in the 
bracket to zero. It may be noted that the same cone is the envelope of 
the planes 



165. Application to the Plane Section. 

We next study the chord joining a pair of reciprocal points on the cubic 
of 160. Take any point in the plane of the section ; then, as we have just 
seen, a cone of screws can be drawn through this point, each ray of which 
crosses two reciprocal screws. This cone is cut by the plane of section in 
two lines, and, accordingly, we see that through any point in the plane of 
section two chords can be drawn through a pair of reciprocal points. The 
actual situation of these chords is found by drawing a pair of tangents to a 
certain hyperbola. This will now be proved. 

The values of 6 and 6 , which correspond to a pair of reciprocal points, 
fulfil the condition 

tan0tan0 =#; 

whence, 

cos (6 -0 ) = \ cos (0 + ) ; 

where, for brevity, we write X instead of 

l+H 
l-H 

If, further, we make + 6 = ty, we shall find, for the equation of the chord, 

Px + Qy + Rz = 0; 
in which, 



ra ,, . m . m 



P = -^ (X + cos 2a) -f sin 2a sin 2i/r + -- (\ + cos 2a) cos 2^, 

22 ^ 

Q = /t sin y3 + cos ft sin 2a ( (\ + cos 2a) sin 2\fr 

A & 



-~ cos 8 sin 2a cos 

7)1- 



= -h" tan 8 - (\ 2 - 1 ) cot B + \hm sin 2-f - ~ (\ - 1) cot B cos 

11-2 



164 THE THEORY OF SCREWS. [165 

The envelope of this chord is found to be 
= +# 2 ra 2 sin 2 2a, 

+ y 2 | ra 2 cos 2 & cos 2 2a - hm sin cos sin 2a - A 2 sin 2 /3 

+ |m 2 X cos 2a cos 2 /3 + |ra 2 X 2 cos 2 /3, 
w 2 

+ x y \ a s i n 2 cos 2 cos ~ m ^ cos ^ a s ^ n & 



| m 2 X sin 2a cos /3 ra&X sin /3, 
+ ac + m/t 2 cos 2a tan fS + hm 2 X sin 2a + mA 2 X tan /3, 
+ y | - w 2 A cos /3 + w/i 2 sin y3 sin 2a + 2/i 3 sin tan /3 - w 2 AX cos cos 2a, 
+ | m 2 A 2 - A 4 tan 2 0. 

Using the data already assumed in 160, and, with the addition now 
made of taking X to be , the equation reduces to 

122a; 2 - 2(% 2 - 16 Ley - 2417^ - 5003y + 245436 = ; 
which, for convenience of calculation, I change into 

x = 1-6 + 21 6 sec <J> 47 5 tan </>, 
y = - 12-8 + 32-8 sec </>. 

This hyperbola has been plotted down in Fig. 36. It obviously touches the 
cubic at three points. I had not anticipated this until the curves were care 
fully drawn ; but, when the theorem was suggested in this manner, it was 
easy to provide the following demonstration : 

The cone of reciprocal chords drawn through any point P breaks up 
into a pair of planes when P lies on the cylindroid. (I use the expression, 
reciprocal chord, to signify the transversal drawn across a pair of reciprocal 
screws on the cylindroid. This is very different from a screw reciprocal to 
the cylindroid.) For, take the screw reciprocal to that which passes 
through P. Then the plane X, through P and this screw, is obviously one 
part of the locus. Draw through P any transversal across a pair of reci 
procals on the cylindroid, then the plane Y, through the centre and this 
transversal, will be the other part of the locus. This pair of planes, X 
and Y, intersect in a ray which we shall call 8. 

A plane of section through P will, of course, usually cut the two planes 
in two rays, and these will be the two reciprocal chords through P. But 
suppose the plane of section happened to pass through 8, then there will be 
only one reciprocal chord through P, and this will, of course, be 8. Now, 
8 must be a tangent to the cylindroid at P. Every chord through P, in 
the plane of Y, must cut the surface again in a pair of reciprocal points. To 
this 8 must be no exception, and as it lies in X, it intersects the screw 



165] THE GEOMETRY OF THE CYLINDROID. 165 

reciprocal to that through P ; therefore the third intersection of S with the 
surface must coalesce with P, or, in other words, S must be a tangent to the 
surface at P. We have thus shown that in the case where two reciprocal 
chords through a point on the cubic coalesce into one, that one must be the 
tangent to the cubic at P. 

But the two reciprocal chords through a point will only coalesce when 
the point lies on the hyperbola, in which case the two chords unite into the 
tangent to the hyperbola. Consider, then, the case where the hyperbola 
meets the cubic at a point P, inasmuch as P lies on the hyperbola, the two 
chords coalesce into a tangent thereto, but because they do coalesce, this line 
must needs be also a tangent to the cubic ; hence, whenever the hyperbola 
meets the cubic the two curves must have a common tangent. Altogether 
the curves meet in six points, which unite into three pairs, thus giving the 
required triple contact between the hyperbola and the cubic. 

If a constant h be added to all the pitches of the screws on a cylindroid, 
then, as is well known, the screws so altered still represent a possible cylin 
droid ( 18). The variations of h produce no alteration in the cubic section 
of the cylindroid ; but, of course, the hyperbola just considered varies with 
each change of h. In every case, however, it has the triple contact, and 
there is also a fixed tangent which must touch every hyperbola. This is 
the chord joining the two principal screws on the cylindroid ; for, as these 
are reciprocal, notwithstanding any augmentation to the pitches, their chord 
must always touch the hyperbola. The system of hyperbolae, corresponding 
to the variations of h, is thus concisely represented ; they must all touch 
this fixed line, and have triple contact with a fixed curve : that is, they must 
each fulfil four conditions, leaving one more disposable quantity for the 
complete definition of a conic. See Appendix, note 4. 

We write the tangent to the hyperbola or the reciprocal chord in the 
form 

L cos 2i/r + M sin 2-f + N = 0. 

If a pair of values can be found for x and y, which will simultaneously satisfy 

L = 0, M = 0, N = 0, 
then every chord of the type 

L cos 2ir 



must pass through this point. The condition for this is, that the discriminant 
of the hyperbola is zero, and we find the discriminant to be 

{m*h sin (X, 2 - 1) (h tan ft + m sin 2a). 



166 THE THEORY OF SCREWS. [165- 

There are two critical cases in which this expression vanishes. It does so if 

h tan ft + m sin 2a = ; 
where the plane of section is tangential to the cylindroid. 

But we also note that the discriminant will vanish if 

h = 0, 
i.e. if the plane of section passes through the centre of the cylindroid. 

We might have foreseen this from the results of the last article ; for the 
plane of Y is a central section, and the hyperbola has evidently degenerated, 
for all the reciprocal chords, instead of touching an hyperbola, merely pass 
through the common apex P. The case of the central section is therefore 
of special interest. 

166. The Central Section of the Cylindroid. 

By this we mean a section of the surface, special in no other sense, save 
that it passes through the centre of the surface. The equation to the 
central section is ( 160) 

y 3 sin ft cos 2 /3 + yx z sin (3 mx z sin 2a + 2mxy cos ft cos 2a + my- sin 2a cos 2 /3 = 0. 
The chord joining points of equal pitch + 6 is 

x (cos 20 + cos 2a) + y cos ft sin 2a + m cot ft sin 2 20 = 0. 
The apex P through which all reciprocal chords pass is 

_ m (\ 2 - 1) cot ft (\ + cos 2) 
1 + 2X cos 2a 4- A, 2 

_ m (A 2 1) cosec ft . sin 2a 
V ~ 1 + 2X cos 2a + X 2 

and in general the co-ordinates of a point on the cubic are 

x = y cos ft tan (6 a), 
y = m cosec ft sin 26. 

One of these curves may be conveniently drawn to scale, from the 
equations 

as = % tan (0 - 25), 
y = - 66 sin 20. 

The parabola, which is the envelope of equal pitch-chords, would in this 
case have as its equation 




THE CENTRAL SECTION OF THE CYLINDROID. 



To fncf p. 166 



167] THE GEOMETRY OF THE CYLINDROID. 167 

The principal screws on the cylindroid both pass through the double 
point ; the two tangents to the curve at this point must therefore both 
touch the parabola. 

The leading feature of the central section is expressed by the important 
property possessed by the chords joining reciprocal screws. If we add any 
constant to the pitches, then we alter X, and, accordingly, the point P, 
through which all reciprocal chords pass, moves along the curve. 

The tangent to the cubic at P meets the cubic again in the point 
reciprocal to P. Two tangents, real or imaginary, can be drawn from P 
to the cubic touching it in the points 1\, T 2 , respectively: as these must 
each correspond to a screw reciprocal to itself, it follows that T l and T 2 
are the screws of zero pitch. We hence see that the two tangents from any 
point on the cubic touch the cubic in points of equal pitch. 

Let a and ft be two screws, and 7 and 8 another pair of screws, and let 
the two chords, a/9 and 78, intersect again on the cubic. If d and 6 be 
the perpendicular distance and angle between the first pair, and d and 
the corresponding quantities for the second pair, then there must be some 
quantity ay, which, if added to all the pitches on the cyliridroid, will make 
a. and /3 reciprocal, and also 7 and 8 reciprocal. We thus have 

(p a + pp + 2&>) cos d sin 6 = 0, 
(Py + Ps + 2w) cos - d sin 6 = ; 
whence, p a + pp d tan = p y + p& d tan 6 ; 

in other words, for every pair of screws, a and /8, whose chords belong to a 
pencil diverging from a common point on the surface, the expression 

Pa + p? d tan 6 

is a constant. The value of this constant is double the pitch of the screw 
of either of the points of contact of the two tangents from P to the curve. 

167. Section Parallel to the Nodal Line. 

If the node on the cubic be at infinity, the form of equation to the cubic 
hitherto employed will be illusory. The nature of this section must therefore 
be studied in another way, as follows : 

Let the plane cut the two perpendicular screws in A and B. Let I be 
the perpendicular OC from upon G, and let rj be the inclination of this 
perpendicular to the axis of x. Then, taking OA as the new axis of x, in 
which case z will be the new y, we have 

x = I tan (t) 0\ 
y = m sin 20. 



168 THE THEORY OF SCREWS. [167, 

Eliminating 6, and omitting the accents, we have, as the equation of the 
cubic, 

yx 1 + mx 2 sin 2ij + l-y + 2lmx cos 2^ ml 2 sin 2?; = 0. 

The chord joining the two points and & has, as its equation, 
mac cos (0 + ) [cos (2rj - - ) + cos (0 - )] + ly 

- ml sin (2vi-0- ) cos (0 + ) - ml cos (0 - ) sin (0 + ) = 0. 

If this be the chord joining the screws of equal pitch, then 

+ = 0, 
and the equation reduces to 

mas (cos 2?7 + cos 20) + ly ml sin 2^ = 0. 

We thus see that this chord, which in the general section envelops a parabola, 
now passes constantly through the fixed point 

x = 0, 

y = + m sin 2?;. 

This result could have been foreseen ; for, consider that screw on the cylin- 
droid (and there must always be one) normal to the plane which it intersects 
at a point P, any ray in the plane through P is perpendicular to this screw, 
and, therefore, by a well-known property of the cylindroid, must intersect 
the curve again in two points of equal pitch. This point P is, of course, 
the point whose existence we have demonstrated above. 

168. Relation between Two Conjugate Screws of Inertia. 

We have found the relation between a pair of conjugate screws of inertia 
so important in the dynamical part of the theory, that it is worth while to 
investigate the properties of the chord joining two such points in the central 
section. It can readily be shown that this chord must envelop a conic. 
This conic and the point P on the cubic through which all reciprocal chords 
will pass, will enable the impulsive screw, corresponding to any instantaneous 
screw, to be immediately determined. For, draw through any point S that 
tangent to the conic which gives S as one of the two conjugate screws of 
inertia which must lie upon it ; let S be the other conjugate screw ; then 
the chord PS will cut the cubic again in the required impulsive screw. 
The two principal screws of inertia are found by drawing from P that 
tangent to the conic which has not P as one of the two conjugate screws 
of inertia. The two intersections of this tangent, with the cubic, are the 
required principal screws of inertia. 

We can also determine the relation between the impulsive screw and the 
instantaneous screw with regard to any section whatever. We have here 



168] THE GEOMETRY OF THE CYLINDROID. 169 

to consider two conies connected with the cubic, viz. the reciprocal conic, 
which is the envelope of reciprocal chords, and the inertia conic, which is 
the envelope of chords of conjugate screws of inertia. We must provide 
a means of discriminating the two tangents from a point P on the cubic 
to either conic ; any ray, of course, cuts the cubic in three points, of which 
two possess the characteristic relation. If P be one of these two, we may 
call this tangent the odd tangent. The other tangent will have, as its 
significant points, the two remaining intersections ; leaving out P, we can 
then proceed, as follows, to determine the impulsive screw corresponding to 
P as the instantaneous screw : 

Draw the odd tangent from P to the inertia conic, and from the con 
jugate point thus found draw the odd tangent to the reciprocal conic. The 
reciprocal point Q thus found is the impulsive screw corresponding to P as 
the instantaneous screw. 

In general there are four common tangents to the two conies. Of these 
tangents there is only one possessing the property, that the same two of its 
three intersections with the cubic are the correlative points with respect 
to each of the conies. These two intersections are the principal screws of 
inertia. 

To determine the small oscillations we find the potential conic, the 
tangents to which are chords joining two conjugate screws of the potential 
( 100). The two harmonic screws are then to be found on one of the two 
common tangents to the two conies. It can be shown that both the inertia 
conic and the potential conic will, like the reciprocal conic, have triple 
contact with the cubic. 



CHAPTER XIV. 

FREEDOM OF THE THIRD ORDER*. 

169. Introduction. 

The dynamics of a rigid body which has freedom of the third order, 
possesses a special claim to attention, for, included as a particular case, we 
have the celebrated problem of the rotation of a rigid body about a fixed 
point. In the theory of screws the screw system of the third order is 
characterised by the feature that the reciprocal screw system is also of the 
third order, and this is a fertile source of interesting theorems. 

We shall first study the screw system of the third order, and its reciprocal. 
We shall then show how the instantaneous screw, corresponding to a given 
impulsive screw, can be determined for a rigid body whose movements are 
prescribed by any screw system of the third order. We shall also point out 
the three principal screws of inertia, of which the three principal axes are 
only special cases, and we shall determine the kinetic energy acquired by a 
given impulse. Finally, we shall determine the three harmonic screws, and 
we shall apply these principles to the discussion of the small oscillations of 
a rigid body about a fixed point under the influence of gravity. 

A screw system of the first order consists of course of one screw. A 
screw system of the second order consists of all the screws on a certain 
ruled surface (the cylindroid). Ascending one step higher, we find that in 
a screw system of the third order the screws are so numerous that a finite 
number (three) can be drawn through every point in space. In the screw 
system of the fourth order a cone of screws can be drawn through every 
point, while to a screw system of the fifth order belongs a screw of suitable 
pitch on every straight line in space. 

170. Screw System of the Third Order. 

We shall now consider the collocation of the screws in space which 
constitute a screw system of the third order. A free rigid body can receive 

* Transactions of the Royal Irish Academy, Vol. xxv. p. 191 (1871). 



169-172] FREEDOM OF THE THIRD ORDER. 171 

six independent displacements. Its position is, therefore, to be specified by 
six co-ordinates. If, however, the body be so constrained that its six co 
ordinates must always satisfy three equations of condition, there are then 
only three really independent co-ordinates, and any position possible for a 
body so circumstanced may be attained by twists about three fixed screws, 
provided that twists about these screws are permitted by the constraints. 

Let A be an initial position of a rigid body M. Let M be moved from 
A to a closely adjacent position, and let x be the screw by twisting about 
which this movement has been effected ; similarly let y and z be the two 
screws, twists about which would have brought the body from A to two 
other independent positions. We thus have three screws, x, y, z, which com 
pletely specify the circumstances of the body so far as its capacity for 
movement is considered. 

Since M can be twisted about each and all of x, y, z, it must be capable 
of twisting about a doubly infinite number of other screws. For suppose 
that by twists of amplitude x , y , z , the final position V is attained. This 
position could have been reached by twisting about some screw v, so as to 
come from A to V by a single twist. As the ratios of x to y , and z , are 
arbitrary, and as a change in either of these ratios changes v, the number 
of v screws is doubly infinite. 

All the screws of which v is a type form what we call a screw system of 
the third order. We may denote this screw system by the symbol S. 

171. The Reciprocal Screw System. 

A wrench which acts on a screw 77 will not be able to disturb the equili 
brium of M, provided t] be reciprocal to x, y, z. If rj be reciprocal to three 
independent screws of the system S, it will be reciprocal to every screw of S. 
Since rj has thus only three conditions to satisfy in order that it may be 
reciprocal to S, and since five quantities determine a screw, it follows that tj 
may be any one of a doubly infinite number of screws which we may term 
the reciprocal screw system S . Remembering the property of reciprocal 
screws ( 20) we have the following theorem ( 73). 

A body only free to twist about all the screws of S cannot be disturbed 
by a wrench on any screw of S ; and, conversely, a body only free to tAvist 
about the screws of S cannot be disturbed by a wrench on any screw of S. 

The reaction of the constraints by which the freedom is prescribed 
constitutes a wrench on a screw of S . 

172. Distribution of the Screws. 

To present a clear picture of all the movements which the body is 



172 THE THEORY OF SCREWS. [172- 

competent to execute, it will be necessary to examine the mutual connexion 
of the doubly infinite number of screws which form the screw system. It 
will be most convenient in the first place to classify the screws in the 
system according to their pitches ; the first theorem to be proved is as 
follows : 

A II the screws of given pitch + k in a three-system lie upon a hyperboloid 
of which they form one system of generators, while the other system of gene 
rators with the pitch k belong to the reciprocal screw system. 

This is proved as follows : Draw three screws, p, q, r, of pitch + k 
belonging to S. Draw three screws, I, m, n, each of which intersects the 
three screws p, q, r, and attribute to each of I, m, n, a pitch k. Since two 
intersecting screws of equal and opposite pitches are reciprocal, it follows 
that p, q, r, must all be reciprocal to I, m, n. Hence, since the former 
belong to S, the latter must belong to 8 . Every other screw of pitch + k 
intersecting I, m, n, must be reciprocal to S , and must therefore belong to S. 

But the locus of a straight line which intersects three given straight 
lines is a hyperboloid of one sheet, and hence the required theorem has 
been proved. 

173. The Pitch Quadric. 

One member of the family of hyperboloids obtained by varying k presents 
exceptional interest. It is the locus of the screws of zero pitch belonging 
to the screw complex. As this quadric has an important property ( 176) 
besides that of being the locus of the screws of zero pitch, it is desirable 
to denote it by the special phrase pitch quadric. 

We shall now determine the equation of the pitch quadric. Let one of 
the principal axes of the pitch quadric be denoted by x, this will intersect 
the surface in two points through each of which a pair of generators can be 
drawn. One generator of each pair will belong to S, and the other to S . 
Each pair of generators will be parallel to the asymptotes of the section of 
the pitch quadric by the plane containing the remaining principal axes 
y and z. Let /*, v be the two generators belonging to S, then lines bisecting 
internally and externally the angle between two lines in the plane of y and 
z, parallel to /i, v will be two of the principal axes of the pitch quadric. 
Draw the cylindroid (pit). The two screws of zero pitch on the cylindroid 
are equidistant from the centre of the cylindroid, and the two rectangular 
screws of the cylindroid bisect internally and externally the angle between 
the lines parallel to the screws of zero pitch. Hence it follows that the two 
rectangular screws of the cylindroid (pv) must be on the axes of y and z 
of the pitch quadric. We shall denote these screws by ft and 7, and their 



174] FREEDOM OF THE THIRD ORDER. 173 

pitches by p$ and p y . From the properties of the cylindroid it appears 
that a, the semiaxis of the pitch quadric, must be determined from the 
equations 

a = (pp py) sin I cos I, 

P cos 2 1 + p y sin 2 1 = ; 
whence eliminating I, we deduce 



with of course similar values of b and c. Substituting these values in the 
equation of the quadric 



we deduce the important result which may be thus stated : 

The three principal axes of the pitch quadric, when furnished with suitable 
pitches p a ,pii, p-f, constitute screws belonging to the screw system of the third 
order, and the equation of the pitch quadric has the form 

PO.CO" + ppf + p y z 2 + p a pppy = 0. 



We can also show conversely that every screw 6 of zero pitch, which 
belongs to the screw system of the third order, must be one of the generators 
of the pitch quadric. For 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 9 must intersect the 
pitch quadric in an infinite number of points, and must therefore be entirely 
contained thereon. 

174. The Family of Quadrics. 

It has been shown that all the screws of given pitch belonging to a 
system of the third order are the generators of a certain hyperboloid. 
There is of course a different hyperboloid for each pitch. We have now 
to show that all these hyperboloids are concentric. 

Take any two screws whatever belonging to the system and draw the 
cylindroid which passes through those screws. This cylindroid contains 
two screws of every pitch. It must therefore have two generators in 
common with every hyperboloid of the family. But from the known sym 
metrical arrangement of the screws of equal pitch on a cylindroid, it follows 
that the centre of that surface must lie at the middle point of the shortest 
distance between each two screws of equal pitch. The centres of the hyper 
boloids for all possible pitches must therefore lie in the principal plane of 
any cylindroid of the system. Take any three cylindroids of the system. 



174 THE THEORY OF SCREWS. [174, 

The centres of all the hyperboloids coincide with the intersection of the 
three principal planes of the cylindroids. It will be convenient to call this 
point the centre of the three-system. 

We hence see that whenever three screws of a three-system are given, 
the centre of the system is determined as the intersection of the principal 
planes of the three cylindroids denned by each pair of screws taken suc 
cessively. 

We may also show that not only are the family of hyperboloids concentric, 
but that they have also their three principal axes coincident in direction and 
situation with the principal axes of the pitch quadric. 

Draw any principal axis z of the pitch quadric. Two screws of zero 
pitch belonging to the system will be intersected by z and we draw the 
cylindroid through these two screws. Let L l and L.> be the two screws of 
equal pitch p on this cylindroid. Let be the centre of the cylindroid, this 
same point being also the centre of the pitch quadric, and therefore as 
shown above of every p-pitch hyperboloid S p . As the centre bisects every 
diameter, it follows that the plane OL 2 cuts the hyperboloid S p again in a 
ray LI which is perpendicular to z and crosses L l at its intersection with z. 
The plane containing L l and Z/ is therefore a tangent to 8 P at the point 
where the plane is cut by z. As z is perpendicular to this plane it follows 
that the diameter is perpendicular to its conjugate plane. Hence z is a 
principal axis of S p , and the required theorem is proved. 

Let now S denote a screw system of the third order, where a, /3, y are 
the three screws of the system on the principal axes of the pitch quadric. 
Dimmish the pitches of all the screws of S 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 ( 110). 

Regarding & as a variable parameter, the equation just written represents 
tlie family of quadrics which constitute the screw system S and the reciprocal 
screw system 8 . Thus all the generators of one system on each quadric, 
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, 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 p a ,pp, p y - 



175] FREEDOM OF THE THIRD ORDER. 175 

Hence the pitches of all the real screws of the screw system S are inter 
mediate between the greatest and least of the three quantities p a> pp, p y . 

175. Construction of a three-system from three given Screws. 

If a family of quadric surfaces have one pair of generators (which do not 
intersect) in common, then the centre of the surface will be limited to a 
certain locus. We may investigate this conveniently by generalizing the 
question into the search for the locus of the pole of a fixed plane with 
respect to the several quadrics. 

Let A be the given plane, / be the ray which joins the two points in 
which the given pair of generators intersect A, X be the plane through / 
and the first generator, Y the plane through / and the second generator, 
B the plane through / which is the harmonic conjugate of A with respect 
to X and Y. Then B is the required locus. 

For, draw any quadric through the two given generators, and let be 
the pole of A with respect to that quadric. 

Draw a transversal through cutting the plane A in the point A l and 
the first and second generators in X l and Y t respectively. Since A 1 is on 
the polar of it follows that OZ^Fj is an harmonic section. But the 
transversal must be cut harmonically by the pencil of planes I(BXAY) 
and hence must lie in B, which proves the theorem. 

In the particular case when A is the plane at infinity, then is the 
centre of the quadric. A plane parallel to the two generators cuts the 
plane at infinity in the line /, and the planes X, Y and B must also contain 
7. Then A, B, X, Y are parallel planes. Any transversal across X and Y 
is cut harmonically by B and A, and as A is at infinity, the transversal must 
be bisected at B. It thus appears that when a family of quadrics have one 
pair of non-intersecting generators in common, then the plane which bisects 
at right angles the shortest distance between these generators is the locus 
of the centres of the quadrics. 

If therefore three generators of a quadric are given, the three planes 
determined by each pair of the quadrics determine the centre by their 
intersection. The construction of the axes of the quadric may be effected 
geometrically in the following manner. Draw three transversals Q l} Q 2 , Q 3 
across the three given generators R 1} R 2 , R 3 . Draw also two other trans 
versals Hi, R 5 across Q 1} Q 2 , Q 3 . Construct the conic which passes through the 
five points in which R 1} R 2 , R 3> R it R s intersect the plane at infinity. Find 
the common conjugate triangle to this conic and to the circle which is the 
intersection of every sphere with the plane at infinity. Then the three 



176 THE THEORY OF SCREWS. [175- 

rays from the centre of the quadric to the vertices of this triangle are the 
three principal axes of the quadric. 

We thus prove again that if a and /3 be any two screws of a three-system, 
the centre of the pitch-quadric must lie in the principal plane of the 
cylindroid through a and /?. For the common perpendicular to any two 
screws of equal pitch on the cylindroid will be bisected by the principal 
plane and therefore any hyperboloid through these two screws of equal 
pitch must have its centre in that plane. 

176. Screws through a Given Point. 

We shall now show that three screws belonging to S, and also three 
screws belonging to 8 , can be drawn through any point x , y , z . Substitute 
x, y , z , in the equation of 17 5 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. 

From the form of the equation it appears that the sum of the pitches of 
three screws through a point is constant and equal to p a +pp + p y - 

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 described around any point as centre, the three screws belonging 
to S, 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 can be found parallel 
to any given direction. All the generators of the quadric are parallel to 

the cone 

(p a - k) x* + (p ft - k) f + (p y - k) z* = 0, 

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 gene 
rators 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 8 has a pitch which is propor 
tional to the inverse square of the parallel diameter of the pitch quadric*. 

* This theorem is connected with the linear geometry ol Plucker, who has shown (Neue Geometric 
des Ratlines, p. 130) that k l x- + L 2 y- + k, t z 2 + k 1 kJ{ 3 ) is the locus of lines common to three 
linear complexes of the first degree. The axes of the three complexes are directed along the 
co-ordinate axes, and the parameters of the complexes are fcj, ._,, k 3 ; the same author has also 
proved that the parameter of any complex belonging to the " dreigliedrige Gruppe" is propor 
tional to the inverse square of the parallel diameter of the hyperboloid. 



176] FREEDOM OF THE THIRD ORDER. 177 

Let r be the intercept on a generator of the cone 

(p a - k) tf + (pp -k)f + (p y -k)z* = 0; 

by the pitch quadric 

p a x" + ptf + W> + paptpy = ; 

then k = -PP^- 

r- 

but k is the pitch of the screw of S, which is parallel to the line r. 

Nine constants ( 75) are required for the determination of a screw 
system of the third order. This is the same number as that required for 
the specification of a quadric surface. We hence infer, what is indeed 
otherwise manifest, viz., that when the pitch quadric is known the entire 
screw system of the third order is determined. 

Another interesting property of the pitch quadric is thus enunciated. 
Any three co-reciprocal screws of a given screw system of the third order 
are parallel to a triad of conjugate diameters of its pitch quadric. 

Take any three co-reciprocal screws of the system as screws of reference, 
and let p lt p^, p 3 be their pitches. If then the co-ordinates of any screw p 
belonging to the system be denoted by p l} p 2 , p 3 , we shall have for the pitch 
ofp(95) 

Pp = PlP* + P*P* 



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 ( 35) 

x y z 

~ = Pi> - = P* -=P*- 

It follows that p p must be proportional to the inverse square of the 
parallel diameter of the quadric surface 

P& 2 + p 2 y n - + p 3 z = H. 

But p ft 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 diameters, and hence three 
conjugate diameters of the pitch quadric are parallel to three co-reciprocal 
screws of the screw system. 

B. 12 



178 



THE THEORY OF SCREWS. 



[176- 



We see from this that the sum of the reciprocals of the pitches of three 
co-reciprocal screws is constant. This theorem will be subsequently 
generalised. 

177. Locus of the feet of perpendiculars on the generators*. 

If p be the pitch of the screw of the three-system which makes angles 
a, ft, 7 with the three principal screws, it is then easy to show that the 
equation of the screw is 

(p a) cos a + z cos /3 y cos 7 = 0, 
z cos a + (p b) cos ft + a; cos 7 = 0, 
+ y cos a x cos ft + (p c) cos 7 = 0. 

If perpendiculars be let fall from the origin on the several screws of the 
system, then if x, y, z be the foot of one of the perpendiculars 

sc cos a + y cos ft + z cos 7 = 0. 

Eliminating cos a, cos /3, cos 7 from this equation and the two last of those 
above, we have 

a; y z = 0, 

z p b x 

+ y x p c 

or (pV)(p-c)x + x (# 2 + y- + z 2 ) + yz (b - c) = ; 

from this and the two similar equations we have, by elimination of p 2 and p 
and denoting x 2 + y z + z* by r 2 , 

x, (b + c) x, bcx + (b c) yz + xr* = ; 
y, (c + a) y, cay + (c a) zx + yr n - 
z, (a+b)z, abz + (a b) xy + zr 3 

multiplying the first column by r- and subtracting it from the last, we have 

x, (b + c) x, bcx + (b c) yz = 0, 
y, (c + a) y, cay + (c a) zx 
z, (a + b) z, abz + (a b) xy 
which may be written 

(a - b) 2 #y + (b - cf y*z- + (c - a) 2 z-ac" = (a -b)(b- c) (c - a) vyz. 

* This Article is due to Professor C. Joly, On the theory of linear vector functions," Transac 
tions of the Royal Irish Academy, Vol. xxx. pp. 601 and 617 (1895), where a profound discussion 
of Steiner s surface is given. See also by the same author Bishop Law s Mathematical Prize 
Examination, Dublin University Examination Papers, 1898. 



178] FREEDOM OF THE THIRD ORDER. 179 

This equation denotes a form of Steiner s surface : 

v a + V/3 + V7 + VS = 0, 
where 



----- __ 

b c c a a o 

Q_ ^ 2y 2z 

r-* 7 7 "T" - j 

6 c c a a b 

2x 2y 2z 

7 = - 7 --- + ^ ------ -, + 1, 

o c c a a o 



.__ -- . 

6 c c a a 6 

From the form of its equation it appears that this surface has three 
double lines, which meet in a point, viz. the three axes OX, OY, OZ. This 
being so any plane will cut the surface in a quartic curve with three double 
points, being those in which the plane cuts the axes. If the plane touch the 
surface, the point of contact is an additional double point on the section, that 
is, the section will be a quartic curve with four double points, i.e. a pair of 
conies. The projections of the origin on the generators of any cylindroid 
belonging to the system lie on a plane ellipse ( 23). This ellipse must lie 
on the Steiner quartic. Hence the plane of the ellipse must cut the quartic 
in two conies and must be a tangent plane. See note on p. 182. 

178. Screws of the Three-System parallel to a Plane. 

Up to the present we have been analysing the screw system 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 is in general 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 system of the second order 
( 72) 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 system which can lie in a given plane 
is, therefore, equal to the number of the quadrics of the system which can 
be drawn to touch that plane. 

122 



180 THE THEORY OF SCREWS. [178, 

The quadric surface whose equation is 
(p a - k) x- + (pp - k) y- + (p y - k) z- + (p* - k) (p? - k) (p y - fc) = 0, 

touches the plane Px + Qy + Rz + S = 0, when the following condition is 
satisfied : 



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 
system, and, of course, two reciprocal screws, lie in the plane. 

From this it follows that all the screws of the system parallel to a plane 
must in general 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 system; but this plane cannot contain any other screws; 
therefore, all the screws parallel to a given plane must lie upon the same 
cylindroid. 

179. Determination of a Cylindroid. 

We now propose to solve the following problem : Given a plane, deter 
mine the cylindroid which contains all the screws, selected from a screw 
system of the third order, which are parallel to that plane. 

Draw through 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 ( 174). 




Fig. 37. 



Let T l} T 2 (Fig. 37) be two points in which the two quadrics of constant 
pitch touch the plane of the paper, which may be regarded as any plane 
parallel to A ; then P is the intersection of the pair of screws belonging 
to the system PT l} PT 2 , which lie in that plane, and P is the intersection 
of the pair of reciprocal screws P R lt P R belonging to the reciprocal 



179] 



FREEDOM OF THE THIRD ORDER. 



181 



system. Since P R^ is to be reciprocal to PT 2 , it is essential that ^ be 
a right angle; similarly ,R 2 is a right angle. The reciprocal cylindroid, whose 
axis passes through P , will be identical with the cylindroid belonging 
to the system whose axis passes through P ; but the two will be differently 
posited. If the angle at P be a right angle, the points T l and T 2 are at 
infinity ; therefore, the plane touches the quadrics at infinity ; it must, 
therefore, touch the asymptotic cone, and must, therefore, pass through the 
centre of the pitch quadric ; 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 a diameter of the pitch quadric 
conjugate to the plane A. Let this line intersect 
the pitch quadric in the points P 1} P 2 , and let S, 
S (Fig. 38) be the feet of the perpendiculars let 
fall from P 1} P 2 upon the plane A. Draw the 
asymptotes OL, OM to the section of the pitch 
quadric, made by the plane A. Through S and S 
draw lines in the plane A, ST, ST , S T, S T , 
parallel to the asymptotes, then T and T are the 
centres of the two required cylindroids which belong 
to the two reciprocal screw systems. 




This construction is thus demonstrated : 



Fig. 38. 



The tangent planes at P^ P 2 each intersect the surface in lines parallel 
to OL, OM. Let us call these lines PI-//I, P\Mi through the point P 1} and 
P.,L,, P Z M., through the point P. 2 . Then P^, PM< are screws belonging 
to the system, and P l M l , P. 2 L 2 are reciprocal screws. 

Since OL is a tangent to the pitch quadric, it must pass through the 
intersection of two rectilinear generators, which both lie in a plane which 
contains OL ; but since OL touches the pitch quadric at infinity, the 
two generators in question must be parallel to OL, and therefore their 
projections on the plane of A must be S T, ST . Similarly for ST, 
S T ; hence ST and S T are the projections of two screws belonging to 
the system, 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 section 
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 



182 THE THEORY OF SCREWS. [179- 

drawn through T parallel to the external and internal bisectors of the angle 
between the asymptotes are the two rectangular 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. 

We may also note that a screw of the system perpendicular to the plane 
passes through T. Thus given any cylindroid of the system the position of 
the screw of the system parallel to the axis of the cylindroid is determined*. 

180. Miscellaneous Remarks. 

We are now in a position to determine the actual situation of a screw 
belonging to a screw system of the third order of which the direction is 
given. The construction is as follows : Draw through the centre of the 
pitch quadric a radius vector OR parallel to the given direction of 6, 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 0. For 
the section in which A cuts the pitch quadric has for a tangent at .R a 
line perpendicular to OR; hence the line OR is a principal axis of the 
section, and hence (179) one of the two screws of the system in the plane 
A must be parallel to OR. It remains to find the actual situation of 6 in 
the plane A. 

Since the direction of is known, its pitch is determinate, 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 6. This quadric must be intersected by the plane A in two parallel 

* In a letter (10 April 1899) Professor C. Joly writes as follows : Any plane through the 
origin contains one pair of screws A and B belonging to the system intersecting at right angles 
and another pair A and B belonging to the reciprocal system. The group A, B, A , B form 
a rectangle of which the origin is the centre. The feet of the perpendiculars from on A and 
on B and the point of intersection of A and B will lie on the Steiner s quartic 

(b-c)-y-z z + (c-a) 2 z-x- + (a-b) 2 x z y 2 - +(b-c) (c-a) (a-b)xyz. 

The point of intersection of A and B and the feet of the perpendiculars on A and B will lie on 
the new Steiner s quartic 

(6 - c) 2 ?/ 2 z 2 + (c - ) z*-x*+(a - &) 2 zy= - (b - c) (c - a) (a - b) xyz. 

The locus of the feet of the perpendiculars on the screws of a three-system from any arbitrary 
origin whatever is still a Steiner s quartic, but its three double lines are no longer mutually rect 
angular. They are coincident with the three screws of the reciprocal three-system which passed 
through the origin. This quartic is likewise the locus of the intersection of the pairs of screws 
of the reciprocal system which are coplanar with the origin. There is a second Steiner s quartic 
whose double lines coincide with the three screws of the given system which pass through the 
origin and which is the locus of intersection of those pairs of screws of the given system which 
lie in planes through the origin. It is also the locus of the feet of perpendiculars on the screws 
of the reciprocal system. 



181] FREEDOM OF THE THIRD ORDER. 183 

lines. One of these lines is the required residence 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. 

The 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 system parallel 
to one of the planes of circular 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 a principal axis 
of the pitch quadric, through each of which a plane pencil of screws can be 
drawn, which belong to the screw system. All the screws passing through 
either of these points have equal pitch. The pitches of the two pencils are 
equal in magnitude, but opposite in sign. The magnitude is that of the 
pitch of the screw situated on the principal axis of the pitch quadric*. 

181. Virtual Coefficients. 

Let p be a screw of the screw system which makes angles whose cosines 
are /, a, 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 p, 



&c., &c., 

ps =/e +g@6 + hy6 
Let ij be any given screw. The virtual coefficient of p and rj is 



Draw from the centre of the pitch quadric a radius vector r parallel to p, 
and equal to the virtual coefficient just written ; then the locus of the 
extremity of r is the sphere 

x 2 + \f + z* = #CT ar) + yet ft + zvr yrl . 

The tangent plane to the sphere obtained by equating the right-hand 
side of this equation to zero is the principal plane of that cylindroid which 
contains all the screws of the screw system which are reciprocal to 17. 

* If a, b, e be the three semiaxes of the pitch quadric, and +d the distances from the centre, 
on a, of the two points in question, it appears from 179 that 2 d 2 = (a 8 -i 2 ) (a 2 -c 2 ), which shows 
that d is the fourth proportional to the primary semiaxis of the surface, and to those of its focal 
ellipse and hyperbola. 



THE THEORY OF SCREWS. [182- 

182. Four Screws of the Screw System. 

Take any four screws a, ft, 7, 8 of the screw system of the third order. 
Then we shall prove that the cylindroid (a, ft) must have a screw in common 
with the cylindroid (7, &). For twists of appropriate amplitudes about a, 
ft, 7, B must neutralise, and hence the twists about a, ft must be counter 
acted by those about 7, & ; but this cannot be the case unless there is 
some screw common to the cylindroids (a, ft) arid (7, 8). 

This theorem provides a convenient test as to whether four screws 
belong to a screw system of the third order. 

183. Geometrical notes. 

The following theorem may be noted : 

Any ray 77 which crosses at right angles two screws a, ft of a three-system 
is the seat of a screw reciprocal to the system. 

For, draw the cylindroid a, ft, then of course 77, whatever be its pitch, 
is reciprocal to all the screws on this cylindroid. Through any point P on 
77 there are two screws of the system which lie on the cylindroid, and there 
must be a third screw 7 of the system through P, which, certainly, does 
not lie on the cylindroid. If, therefore, we give 77 a pitch p y) it must be 
reciprocal to the three-system. 

In general, one screw of a three-system can be found which intersects 
at right angles any screw ivhatever 77. 

For 77 must, of course, cut each of the quadrics containing the screws 
of equal pitch in two points. Take, for example, the quadric with screws 
of pitch p. There are, therefore, two screws, a and ft of pitch p belonging 
to the system, which intersect 77. The cylindroid a, ft must belong to the 
system, and from the known property of the cylindroid the ray ij, which 
crosses the two equal pitch screws ( 22), must cross at right angles some 
third screw 7 on this cylindroid ; but this belongs to the three-system, and 
therefore the theorem has been proved. 

184. Cartesian Equation of the Three-System. 

If we are given the co-ordinates of any three screws of a three-system 
with reference to six canonical co -reciprocals, we can calculate in the 
following manner the equation to the family of pitch quadrics of which the 
three-system is constituted. 

Let the three given screws be a, ft, 7, with co-ordinates respectively 



184] 



FREEDOM OF THE THIRD ORDER. 



185 



!,...; ft l} ... /3; 71, ... 7 6 . Then if X, p, v be three variable parameters, 
the co-ordinates of the other screws of the three-system will be 

Xotj + yttySj + vji , Xa. 2 + /i$j + v%, ... Xa 6 + /Jift 6 + i>7 6 . 

We shall denote the pitch of this screw by p, and from 43 we have for the 
equations of this screw with reference to the associated Cartesian axes : 

= + (Xa 5 + pfa + vy, + Xa 6 + fj,j3 9 + vy 6 ) y 

(X 3 + fj,ft s + vj 3 + X 4 -1- pfti + v%) z 

(Xj + jjifii + vyi X 2 /A/3 a 1/72) a 



with two similar equations. 

From these we eliminate X, /*, v and the determinant thus arising admits 
of an important reduction. 

To effect this we multiply it by the determinant 



4 , 



. 



I 7i + 7a . 73 + 74 
For brevity we introduce the following notation : 

P = x [(ft, + &) (73 + 74) - 08, + A) (75 + 7.)] 

+ y [(& + &) ( 75 + 7) - (& + ft) (71 + 7.)] 

+ z [(& + &) ( 7l + 7.) - (^ + /3 2 ) (7, + 74 )] , 
with similar values for Q and R by cyclical interchange. 
We also make 

L aft = a(a 1 + a ) (A - &) + 6 ( 3 + 4 ) (/3 3 - /9 4 ) + c (a, + 6 ) (/3 5 - &), 
^ = a (A + A) (i - a,) + & (& + /3 4 ) (a 3 - 4 ) + c (/3 5 + &) (a 8 - 6 ), 
with similar values for Z ay , Z ya , L fty , L^ by cyclical interchange. 

The equation to the family of pitch quadrics is then easily seen to be 
0= 



If the three given screws a, ft, 7 had been co-reciprocal, then as 

L a p + Lp a = 2-57 a = 0, 
it follows that L af} and L fta only differ in sign, so that if 



186 THE THEORY OF SCREWS. [184- 

the equation becomes 

= p a -p , R p cos (a/3), + Q p cos (017) 

R -p cos (aft), Pfi-p 

- Q - P cos ( a 7) > + P - p cos 

By expanding this as a cubic for p we see that the coefficient of p 2 
divided by that of p 3 with its sign changed is 

p a sin 2 (ffy) +jp/3 sin 2 (yq) + jo Y sin 2 (off) 

sii4[(y) + ( 7 a) + (a/3)]sin|[^ 

This is accordingly the constant sum of the three pitches of the screws of 
the system which can be drawn through any point. 



185. Equilibrium of Four Forces applied to a Rigid Body. 

If the body be free, the four forces must be four wrenches on screws of 
zero pitch which are members of a screw system of the third order. The 
forces must therefore be generators of a hyperboloid, all belonging to the 
same system ( 132). 

Three of the forces, P, Q, R, being given in position, S must then be a 
generator of the hyperboloid determined by P, Q, R. This proof of a 
well-known theorem (due to Mobius) is given to show the facility with 
which such results flow from the Theory of Screws. 

Suppose, however, that the body have only freedom of the fifth order, 
we shall find that somewhat more latitude exists with reference to the 
choice of S. Let X be the screw reciprocal to the screw system by which 
the freedom is defined. Then for equilibrium it will only be necessary that 
S belong to the system of the fourth order defined by the four screws 

P, Q, R, X. 

A cone of screws can be drawn through every point in space belonging 
to this system, and on that cone one screw of zero pitch can always be 
found ( 123). 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 latitude in the choice 
of S is still greater. Let X ly X 2 be two screws reciprocal to the system, 
then S is only restrained by the condition that it belong to the screw system 
of the fifth order defined by the screws 

P, Q, R, X 1} X. 



186] FREEDOM OF THE THIRD ORDER. 187 

Any line in space when it receives the proper pitch is a screw of this 
system. 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 
system ( 110). 

Finally, if the body has only freedom of the third order, the four equi 
librating forces P, Q, R, S may be situated anywhere. 

The positions of the forces being given, their magnitudes are determined ; 
for draw three screws X lt X. 2> X 3 reciprocal to the system, and find ( 28) the 
intensities of the seven equilibrating wrenches on 

4j Q> -R> >> Xj, X 2 , X 3 . 

The last three are neutralised by the reactions of the constraints, and 
the four former must therefore equilibrate. 

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 are destroyed by the reactions, and, therefore, 
the four wrenches on the four screws equilibrate. It is manifest that this 
theorem may be generalised into the following : If a body have freedom of 
the kth order, then properly selected wrenches about any k+l screws (not 
reciprocal to the screw system) 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 system ; 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. 

186. The Ellipsoid of Inertia. 

The momental ellipsoid, 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 
system of the third order. 



188 THE THEORY OF SCREWS. [186- 

If we take three conjugate screws of inertia from the screw system as 
screws of reference, then we have seen (97) that, if 1} 0.,, 3 , be the co 
ordinates of a screw 0, we have 



where u lt u. 2 , u s are the values of u g with reference to the three conjugate 
screws of inertia. 

Draw from any point lines parallel to 0, and to the three conj ugate screws 
of inertia. If then a parallelepiped be constructed of which the diagonal is 
the line parallel to 0, and of which the three lines parallel to the conjugate 
screws are conterminous edges, and if r be the length of the diagonal, and 
x, y, z the lengths of the edges, then we have 

x _a V a z a 
r~ * IT = 2> r if* 

We see, therefore, that the parameter u appropriate to any screw is 
inversely proportional to the parallel diameter of the ellipsoid 



u.?z> = H, 
where H is a 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 system 
of the third order, is proportional to the inverse square of the parallel 
diameter of a certain ellipsoid, which may be called the ellipsoid of inertia ; 
and a set of three conjugate diameters of the ellipsoid are parallel to a set 
of three conjugate screws of inertia which belong to the screw system. 

We might also enunciate the property in the following 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 system, so 
that its kinetic energy shall be constant. 

187. 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. W T e can then determine 
three screws of the system parallel to these diameters ( 180), and these 
three screws will be co-reciprocal, and also conjugate screws of inertia. 
They will, therefore, ( 87), form what we have termed the principal screws 
of inertia. When the screw system 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. 



190] FREEDOM OF THE THIRD ORDER. 189 

188. Lemma. 

If from a screw system of the nih order we select n screws A ly ... , A n , 
which are conjugate screws of inertia ( 87), and if 8 l be any screw which 
is reciprocal to A 2 ,...,A n , then an impulsive wrench on S Y will cause the 
body, when only free to twist about the screws of the system, to commence 
to twist about A lt Let 7^ be the screw which, if the body were perfectly 
free, would be the impulsive screw corresponding to A l as the instantaneous 
screw. R! must be reciprocal to A.,,...,A n ( 81). Take also 6 n screws 
of the reciprocal system B 1 ,...,B ti ^ n . Then the 8 n screws R l} S lt B l} ... , 
B 6 _ n must be reciprocal to the n 1 screws A 2 , ...,A n , and therefore the 
8 n screws must belong to a screw system of the (7 ?i)th order. Hence 
an impulsive wrench upon the screw Si can be resolved into components on 
RI, BI, ... ,B 6 - n . Of these all but the first are neutralised by the reactions 
of the constraints, and by hypothesis the effect of an impulsive wrench 
upon Q is to make the body commence to twist about A 1} and therefore 
an impulsive wrench on Si would make the body twist about AI. 

189. Relation between the Impulsive Screw and the Instan 
taneous Screw. 

A quiescent rigid body which possesses freedom of the third order is 
acted upon by an impulsive wrench about a given screw 77. It is required 
to determine the instantaneous screw 6, about which the body will commence 
to twist. 

The screws which belong to the system, and are at the same time reci 
procal to 77, must all lie upon a cylindroid, as they each fulfil the condition 
of being reciprocal 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 77. 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 fi and v denote two screws of the system parallel to a pair of 
conjugate diameters of the ellipsoid of inertia in the reciprocal plane. Then 
6, fi, v are a triad of conjugate screws of inertia ; but 77 is reciprocal to //, 
and v, and, therefore, by the lemma of the last article, an impulsive wrench 
upon i] will make the body commence to twist about 6. 

190. Kinetic Energy acquired by an Impulse. 

We shall now consider the following problem : A quiescent rigid body 
of mass M receives an impulsive wrench of intensity rf" on a screw 77. We 
have now to determine the locus of a screw belonging to a screw system 
of the third order, such that, if the body be constrained to twist about 6, it 



190 THE THEORY OF SCREWS. [190- 

shall acquire a given kinetic energy E, in consequence of the impulsive 
wrench. 

We have from 91 the equation 

1 7/" 2 

We can assign a geometrical interpretation to this equation, which will 
lead to some interesting results. 

Through the centre of the pitch quadric the plane A reciprocal to 77 
is to be drawn. A sphere (181) is to be described touching the plane A 
at the origin 0, the diameter of the sphere being so chosen that the intercept 
OP made by the sphere on a radius vector parallel to any screw 6 is equal 
to ty^g (181). The quantity u e is inversely proportional to the radius vector 
OQ of the ellipsoid of inertia, which is parallel to 9 ( 186). Hence for all 
the screws of the screw system which acquire a given kinetic energy in 
consequence of a given impulse, we must have the product OP . OQ constant. 

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, 0. 

Since we have already shown how, when the direction of a screw belonging 
to a screw system of the third order is given, the actual situation of that 
screw is determined ( 180), we are now enabled to ascertain all the screws 
6 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 proportional to OP . OQ, 
and therefore to the square root of E. 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, therefore, degenerate to the single screw parallel to the diameter 
of the ellipsoid of inertia conjugate to A, But we have seen ( 130) 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 system of the 
third order. We thus see again what Euler s theorem ( 94) would have 
also told us, viz., that when a quiescent rigid body which has freedom of the 
third order is set in motion by the action of a given 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 system than that 
one which it naturally chooses. 



192] FREEDOM OF THE THIRD ORDER. 191 

191. Reaction of the Constraints. 

An impulsive wrench on a screw 77 acts upon a body with freedom of 
the third order, and the body commences to move by twisting upon a screw 
6. It is required to find the screw X, 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 6 as the instantaneous screw. 
Then \ must lie upon the cylindroid (</>, 77), and may be determined by 
choosing on (<, 77) a screw reciprocal to any screw of the given screw 
system. 

192. Impulsive Screw is Indeterminate. 

Being given the instantaneous screw 6 in a system of the third order, 
the corresponding impulsive screw 77 is indeterminate, because the impulsive 
wrench may be compounded with any reactions of the constraints. In fact 
77 may be any screw selected from a screw system of the fourth order, which 
is thus found. Draw the diametral plane conjugate to a line parallel to 6 
in the ellipsoid of inertia, and construct the cylindroid which consists of 
screws belonging to the screw system parallel to this diametral plane. 
Then any screw which is reciprocal to this cylindroid will be an impulsive 
screw corresponding to 6 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 given screw of the screw system. For a 
couple in a plane perpendicular to the nodal line of a cylindroid may be 
regarded as a wrench upon a screw reciprocal to the cylindroid ; and hence 
a couple in a diametral plane of the ellipsoid of inertia, conjugate to the 
diameter parallel to the screw 6, will make the body commence to twist 
about the screw 6. 

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 
impulsive 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. 



192 THE THEORY OF SCREWS. [193- 

193. Quadric of the Potential. 

A body which has freedom of the third order is in equilibrium under 
the influence of a conservative system of forces. The body receives a twist 
of small amplitude 9 about a screw 9 of the screw system. It is required 
to determine a geometrical representation for the quantity of work which 
has been done in effecting the displacement. We have seen that to each 
screw 9 corresponds a certain linear parameter V B ( 102), and that the work 
done is represented by 

F^&\ 

We have also seen that the quantity v e ~ may be represented by 



where l} 2 , 9 3 are the co-ordinates of the screw 9 referred to three conjugate 
screws of the potential, and v lt v. 2 , v s , denote the values of v g for each of the 
three screws of reference ( 102). 

Drawing through the centre of the pitch quadric three axes parallel to 
the three screws of reference, we can then construct the quadric of which 
the equation is 

v^x z + v?y- + v 3 2 z- = H, 

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 system of the third order, is 
proportional to the inverse square of the parallel diameter of a certain 
quadric which we may call the quadric of the potential, and three conjugate 
diameters of this quadric are parallel to three conjugate screws of the 
potential in the screw system. 

194. The Principal Screws of the Potential. 

The three common conjugate diameters of the pitch hyperboloid, and 
the quadric of the potential, are parallel to three screws of the system 
which we call the principal 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 system 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 minimum consumption of 
energy, and they are rectangular, which, of course, is not in general the 
case with the principal screws of the potential. 



196] FREEDOM OF THE THIRD ORDER. 193 

195. Wrench evoked by Displacement. 

By the aid of the quadric 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 screw 6 from a position of stable equilibrium. The 
construction which will now be given will enable us to determine the screw 
of the system on which the reduced wrench acts. 

Draw through the centre of the pitch quadric a line parallel to 6. Con 
struct the diametral plane A of the quadric of the potential conjugate to 
this line, and let X, p, be any two screws of the system parallel to a pair of 
conjugate diameters of the quadric of the potential 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 <f> will then be reciprocal to both X and //,; and as X, /A, 6 are 
conjugate screws of the potential, it follows that a twist about 6 must evoke 
a reduced wrench on <. 

196. Harmonic Screws. 

When a rigid body has freedom of the third order, it must have ( 106) 
three harmonic screws, or screws which are conjugate screws of inertia, 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 system 
which are parallel to the triad of conjugate diameters common to the ellipsoid 
of inertia, and the quadric of the potential. 

We have thus a complete geometrical conception of the small oscillations 
of a rigid body which has freedom 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 harmonic screws. 

If the displacement of the body from its position of equilibrium has 
been effected by a small twist about a screw on the cylindroid which contains 
two of the harmonic screws, then the twist can be decomposed into com 
ponents on the harmonic screws, and the instantaneous 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 system would be a harmonic screw. 

B. 13 



194 THE THEORY OF SCREWS. [197 

197. 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, is 
in stable equilibrium under the influence of gravity. The body is slightly 
disturbed : it is required to determine its small oscillations. 

Since three co-ordinates are required to specify the position of a body 
when rotating about a point, it follows that the body has freedom of the 
third order. The screw system, however, assumes a very extreme type, 
for the pitch quadric has become illusory, and the screw system reduces to 
a pencil of screws of zero pitch radiating in all directions from the fixed 
point. 

The quantity u e 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 present case to the well-known momental ellipsoid. 

The quadric of the potential ( 193) assumes a remarkable form in the 
present case. The work done in giving the body a small twist is propor 
tional to the vertical distance through which the centre of inertia is 
elevated. In the position of equilibrium the centre of inertia is vertically 
beneath the point of suspension, it is therefore 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 cylinder 
must be infinite, because no work is done in rotating the body around a 
vertical axis. 

Let be the centre of suspension, and 1 the centre of inertia, and let 
OP be a radius vector of the quadric of the potential. Let fall 1Q per 
pendicular on OP, and PT perpendicular upon 01, It is extremely easy 
to show that the vertical height through which / is raised is proportional 
to IQ 2 x OP 2 ; whence the area of the triangle OPI is constant, and there 
fore the locus of P must be a right circular cylinder of which 01 is the 
axis. 

We have now to find the triad of conjugate diameters common to the 
momental ellipsoid, and the circular cylinder just described. A group of 
three conjugate diameters 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 conjugate 

* Trans. Roy. Irish Acad., Vol. xxiv, Science, p. 593 (1870). 



197] FREEDOM OF THE THIRD ORDER. 195 

diameters common to 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. 

As to that vertical axis which appears to be one of the harmonic 
axes, the time of vibration about it would be infinite. The three har 
monic 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 
0, conjugate to the vertical diameter 01 of the momental ellipsoid, then the 
common conjugate diameters of the two ellipses in which 8 cuts the momental 
ellipsoid, and a circular cylinder whose axis is 01, are the two harmonic axes. 
If the body be displaced by a small rotation 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 circumstances of the rigid body, 
a few additional remarks are necessary. 

Assuming the body in any given position of equilibrium, it is first to be 
displaced by a small rotation about an axis OX. Draw the plane containing 
01 and OX, and let it cut the plane S in the line OF. The small rotation 
around OX may be produced by a small rotation about 01, followed by a 
small rotation about OF. The effect of the small rotation about 01 is 
merely to alter the azimuth of the position, but not to disturb the equi 
librium. Had we chosen this altered position as that position of equilibrium 
from which we started, the initial displacement would be communicated by a 
rotation around F. We may, therefore, without any sacrifice of generality, 
assume that the axis about which the initial displacement is imparted lies 
in the plane S. We shall now suppose the body to receive a small angular 
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 into components about the two harmonic axes, will have a 
component around the vertical axis 01. An initial rotation about 01 would 
give the body a continuous rotation around the vertical axis, which is 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 instantaneous axis in that plane. The initial displacement may 
be resolved into two displacements, one on each of the harmonic axes, and 

132 



196 THE THEORY OF SCREWS. [197 

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 instan 
taneous 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 con 
jugate diameters of the projected ellipse, it follows that the harmonic axes 
must project 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. 



CHAPTER XV. 

THE PLANE REPRESENTATION OF FREEDOM OF THE THIRD ORDER*. 

198. A Fundamental Consideration. 

Let x, y, z denote the Cartesian co-ordinates of a point in the body 
referred to axes fixed in space. When the body moves into an adjacent 
position these co-ordinates become, respectively, x + 8x, y + 8y, z + 82, and 
we have, by a well-known consequence of the rigidity of the body, 

Bx = a + gz hy, 
8y = b + hx fz, 
8z = c+fy- gx, 

where a, b, c, /, g, h may be regarded as expressing the six generalized 
co-ordinates of the twist which the body has received. 

If the body has only three degrees of freedom, its position must be 
capable of specification by three independent co-ordinates, which we shall 
call 6 lt 6.,, 6. A . The six quantities, a, b, c, f, g, h, must each be a function 
of #!, 2 , 3 , so that when the latter are given the former are determined. 
As all the movements are infinitely small, it is evident that these equations 
must in general be linear, and of the type 

a = A 1 1 + A,0, + A 3 0. J , 

in Avhich A lt A 2 , A 3 are constants depending on the character of the 
constraints. . We should similarly have 

b = B 1 1 + B& + B 3 6 3 , 
and so on for all the others. 

It is a well-known theorem that the new position of the body defined 
by 0i, 0-2, 3 may be obtained by a twist about a screw of which the axis 
is defined by the equations 

a + gz hy _ b + hx fz _ c +fy gx 
~7~ 9 & 

* Trans. Roy. Irish Acad., Vol. xxix. p. 247 (1888). 



198 THE THEORY OF SCREWS. [198- 

The angle through which the body has been rotated is 

(/ 2 + <7 2 + /* 2 )", 
and the distance of translation is 

af+ bg + ch 



while the pitch of the screw is 

af+bg + ch 



Every distinct set of three quantities, d l , #,, 3 , will correspond to a 
definite position of the rigid body, and to a group of such sets there will be 
a corresponding group of positions. Let p denote a variable parameter, and 
let us consider the variations of the set, 



according as p varies. To each value of p a corresponding position of the 
rigid body is appropriate, and we thus have the change of p associated with 
a definite progress of the body through a series of positions. We can give 
geometrical precision to a description of this movement. The equations to 
the axis of the screw, as well as the expression of its pitch, only involve the 
ratios of a, b, c,f, g, h. We have also seen that these quantities are each 
linear and homogeneous functions of l , #,, 3 . If, therefore, we substitute 
for #j, # 2 , # 3 the more general values 



the screw would remain unaltered, both in position and in pitch, though 
the angle of rotation and the distance of translation will each contain p 
as a factor. 

Thus we demonstrate that the several positions denoted by the set p6 l} 
p0. 2 , pO s are all occupied in succession as we twist the body continuously 
around one particular screw. 

199. The Plane Representation. 

All possible positions of the body correspond to the triply infinite triad 



If, for the moment, we regard these three quantities as the co-ordinates 
of a point in space, then every point of space will be correlated to a position 
of the rigid body. We shall now sort out the triply infinite multitude of 
positions into a doubly infinite number of sets each containing a singly 
infinite number of positions. 



200] PLANE REPRESENTATION OF THE THIRD ORDER. 190 

If we fix our glance upon the screws about which the body is free to 
twist, the principle of classification will be obvious. Take an arbitrary triad 

01, 2, 03, 

and then form the infinite group of triads 



for every value of p from zero up to any finite magnitude : all these triads 
will correspond to the positions attainable by twisting about a single screw. 
We may therefore regard 

ly 0. 2 , 6 3 

as the co-ordinates of a screw, it being understood that only the ratios of 
these quantities are significant. 

We are already familiar with a set of three quantities of this nature 
in the well-known trilinear co-ordinates of a point in a plane. We thus 
see that the several screws about which a body with three degrees of 
freedom can be twisted correspond, severally, with the points of a plane. 
Each of the points in a plane corresponds to a perfectly distinct screw, 
about which it is possible for a body with three degrees of freedom to be 
twisted. Accordingly we have, as the result of the foregoing discussion, the 
statement that 

To each screw of a three-system corresponds one point in the plane. 
To develope this correspondence is the object of the present Chapter. 

200. The Cylindroid. 

A twist of amplitude 6 on the screw 6 has for components on the three 
screws of reference 

0i, 0. 2 , S ; 

a twist of amplitude < on some other screw < has the components 



When these two twists are compounded they will unite into a single twist 
upon a screw of which the co-ordinates are proportional to 



If the ratio of to & be X, we see that the twists about and < unite into 
a twist about the screw whose co-ordinates are proportional to 

01 + \<f> ly 6. 2 + X._,, 0-j + \<> s . 



By the principles of trilinear co-ordinates this point lies on the straight line 
joining the points and <. As the ratio \ varies, the corresponding screw 



200 THE THEORY OF SCREWS. [200- 

movcs over the cylindroid and the corresponding point moves over the 
straight line. Hence we obtain the following important result : 

The several screws on a cylindroid correspond to the points on a straight 
line. 

In general two cylindroids have no screw in common. If, however, the two 
cylindroids be each composed of screws taken from the same three-system, 
then they will have one screw in common. This is demonstrated by the 
fact that the two straight lines corresponding to these cylindroids necessarily 
intersect in a point which corresponds to the screw common to the two 
surfaces. 

Three twist velocities about three screws will neutralize and produce 
rest, provided that the three corresponding points lie in a straight line, and 
that the amount of each twist velocity is proportional to the sine of the 
angle between the two non-corresponding screws. 

Three wrenches will equilibrate when the three points corresponding to 
the screws are collinear, and when the intensity of each wrench is propor 
tional to the sine of the angle between the two non -corresponding screws. 

201. The Screws of the Three-system. 

In any three-system there are three principal screws at right angles to 
each other, and intersecting in a point ( 173). It is natural to choose these 
as the screws of reference, and also as the axes for Cartesian co-ordinates. 
The pitches of these screws are p 1} p.,, p 3 , and we shall, as usual, denote the 
screw co-ordinates by 0,, a , 3 . The displacement denoted by this triad of 
co-ordinates is obtained by rotating the body through angles 1} 0. 2 , 3 around 
three axes, und then by translating it through distances p^, p,0. 2 , p 3 3 parallel 
to these axes. As these quantities are all small, we have, for the displace 
ments produced in a point x, y, z, 



Sy = p. 2 2 + x0 3 - z6 l , 
8z = p s 3 + y0 1 -x0 2 ; 

these displacements correspond to a twist about a screw of which the axis 
has the equations 

pA + z0y - y0 3 _ p 2 2 + x0 3 - z0 l p 3 3 + yfl - x0. 2 

0i 0, ~0T~ 

while the pitch p is thus given : 



202] PLANE REPRESENTATION OF THE THIRD ORDER. 201 

We have now to investigate the locus of the screws of given pitch, and as 
p is presumed to be a determinate quantity, we have 



(p.* - p) 2 + x0 3 = 0, 



whence, by eliminating 1} 2 , 3 we obtain, as the locus of the screws of 
pitch p, the quadric otherwise found in the previous chapter 

(p! -p) x z + (p,-p) y + (p.-p) z- + (pi-p)(p s -p)(p 3 -p) = 0. 

According as p varies, this family of quadrics will exhibit all the screws of 
the three-system which possess a definite pitch. 

202. Imaginary Screws. 

To complete the inventory of the screws it is, however, necessary to 
add those of indefinite pitch, i.e. those whose co-ordinates satisfy both the 
equations 

M +M a +M 8 =o. 

0s + e.? + <v=o. 

There are four triads of co-ordinates which satisfy these conditions, and, 
remembering that only the ratios are concerned, the values of 1} n , 3 
may be written thus : 



The equations of the axis written without p are 

* (Of + 0J} - yOA - ^0 A + ( P* - p 3 ) 0,6, = 0, 
y (Of + 0s) - zOA - x6A + (p 3 - Pl ) 6A = o, 

z (6? + <9 2 ) - x0A - y0 3 0, + ( PI - p z ) 6,0, = 0, 
of which two are independent. 

If we substitute the values of 0,, 6.,, 3 for the first indeterminate screw, 
the three equations just written reduce to 

* ( P2 - #,)* + y ( p 3 - p$ + z ( Pl - p$ -(p 2 - p 3 ) h - (p-.-p^^p,- p$ - 0. 



202 THE THEORY OF SCREWS. [202, 

If we make 

* = (P*-pa)* , @ = (p 3 -pi)* , f Y = (pi-prf, 
the equations of the four planes are expressed in the form 

+ ax + fty + yz a/3y = 0, 
ax + fiy + jz afty = 0, 
+ ax /3y + yz a/3y 0, 
+ ax + (3y <yz afty = 0. 

It is remarkable that the three equations of the axis for each of these screws 
here coalesce to a single one. The screw of indeterminate pitch is thus 
limited, not to a line, but to a plane. The same may be said of each of 
the other three screws of indeterminate pitch ; they also are each limited 
to a plane found by giving variety of signs to the radicals in the equations 
just written. We have thus discovered that the complete locus of the 
screws of a three-system consists, not only of the family of quadrics, which 
contain the screws of real or imaginary, but definite pitch, but that it also 
contains a tetrahedron of four imaginary planes, each plane being the locus 
of one of the four screws of indefinite pitch. 

203. Relation of the Four Planes to the Quadrics. 

The planes have an interesting geometrical connexion with the family of 
quadrics, which we shall now develop. The first theorem to be proved is, 
that each of the quadrics touches each of the planes. This is gcometrically 
obvious, inasmuch as each quadric contains all the screws of the system 
which have a given pitch p\ but each of the planes contains a system of 
screws of every pitch, among which there must be one of pitch p. There 
will thus be a ray in the plane, which is also a generator of the hyper- 
boloid but this, of course, requires that the plane be a tangent to the 
hyperboloid. 

It is easy to verify this by direct calculation. 
Write the quadric, 

( Pi - p) a* + (p,- p) f + (p 3 -p)z~ + ( Pl - p) (p. 2 - p) (p 3 - p ) = 0. 

The tangent plane to this, at the point x, y , z, is 
( p, - p) xx + (p,-p) yij + (^ 3 _ p) zz > + ( pl 

If we identify this with the equation 

ax + fiy + ryz a/3y = 0, 



203] PLANE REPRESENTATION OF THE THIltD ORDER. 203 

we shall obtain 



*- Pi)* (Pi- Pi)* 



- (PI-P)(P*-P) 



and as these values satisfy the equation of the quadric, the theorem has been 
proved. 

The family of quadric surfaces are therefore inscribed in a common tetra 
hedron, and they have four common points, as well as four common tangent 
planes. For, write the two cones 

#- + if + z- = 0, 

pj? + ihy" + ihz" = o. 

These cones have four generators in common, and the four points in which 
these generators cut the plane at infinity will lie on every surface of the 
type 

(Pi ~ P) ? + (P* ~ P) V" + (Pa ~p)2- + (pi - p) (p* - p) (p 3 -p) = 0. 

We now see the distribution of the screws in the imaginary planes. In 
each one of these planes there are a system of parallel lines ; each line of this 
system passes through the same point at infinity, which is, of course, one of 
the four points just referred to. Every line of the parallel set, when it 
receives appropriate pitch, belongs to the three-system. 

It thus appears that the ambiguity in the pitches of the screws in the 
planes is only apparent. The system of screw co-ordinates which usually 
defines a screw with absolute definiteness, loses that definiteness for the 
screws in these planes. Each plane contains a whole pencil of screws, 
radiating from a point at infinity, but the co-ordinates can only represent 
these screws collectively, for the three co-ordinates then represent, not a single 
screw, but a whole pencil of screws. As the pitches vary on every screw of 
the pencil, the co-ordinates can only meet this difficulty by representing the 
pitch as indeterminate. 

The proof that only a single screw of each pitch is found in the pencil is 
easily given. If there were two, then the same hyperboloid would have two 
generators in this plane of equal pitch ; but this is impossible, because, from 
the known properties of the three-system, only one of these generators 
belongs to the three-system, and the other to the reciprocal system. 



204 THE THEORY OF SCREWS. [204, 

204. The Pitch Conies. 

The discussion in 203 will prepare us for the plane representation of 
the screws of given pitch p, for we have 

P (0? + 0/ + 3 2 ) - M a - M 2 - M 2 = - 

This, of course, represents a conic section, and, accordingly, we have the 
following theorem : 

The locus of points corresponding to screws of given pitch is a conic 
section. 

A special case is that where the pitch is zero, in which case the locus is 
given by 



This we shall often refer to as the conic of zero -pitch. 

Another important case is that where p is infinite, in which case the 
equation is 

0* + 3 *+0 3 *=o. 

The conic of zero-pitch and the conic of infinite pitch intersect in four points, 
and through these four points all the other conies must pass. The points, of 
course, correspond to the screws of indeterminate pitch : we may call them 
P,, P,, P,, 1\. 

Any conic through these four critical points will be a conic of equal- 
pitch screws. 

As a straight line cuts a conic in two points, we see the well-known 
theorem, that every cylindroid will contain two screws of each pitch. 

The two principal screws on a cylindroid are those of maximum and 
minimum pitch ; they will be found by drawing through P l} P 2 , P 3 , P 4 , the 
two conies touching the straight line corresponding to the cylindroid. The 
two points of contact are the screws required. 

If a and ft are the two principal screws on a cylindroid, then any pair 
of harmonic conjugates to a and ft represent a pair of screws of equal pitch. 

For if S + kS = be a system of conies, then it is well known that the 
pairs of points in which a fixed ray is cut by this system form a system in 
involution. The double points of this involution are the points of contact 
of the two conies of the system which touch the line. 

205. The Angle between Two Screws. 

From the equations of the screw given in 201, we see that the direction 



205] PLANE REPRESENTATION OF THE THIRD ORDER. 205 

cosines are proportional to 8 } , # 2 > ;1 ; for if we take the point infinitely 
distant we find that the equations reduce to 

x _ y _ z 
6 l &, $3 

Accordingly, the line drawn parallel to the screw through the origin has its 
direction cosines proportional to 0,, 2 , 3 , and hence the actual direction 
cosines are 

ft ft 3 



The cosine of the angle between two screws, 6 and </>, will therefore be 



By the aid of the conic of infinite pitch we can give to this a geometrical 
interpretation. 

The co-ordinates of a screw on the straight line joining 6 and <f> will be 

ft + X<j , # 2 + ^-$2> ft + X< 3 . 

If we substitute this in the equation to the conic of infinite pitch we obtain 

ft 2 + ft 2 + 9* -H 2X (#!</>, + 2 </> 2 + 3 3 ) + \- (fa- 4- </>.;- + </) = 0. 
Writing this in the form 

aX 2 + 2&X + c = 0, 

of which Xj and X 2 are the roots, we have, as the four values of X, corre 
sponding, respectively, to the points 6 and </>, and to the points in which 
their chord cuts the conic of infinite pitch, 

X 1( Xo, 0, oo. 

The anharmonic ratio is 

^ 
V 
or 



b - V6 2 - ac 
b + V6 2 - ac 

If to be the angle between the two screws, 6 and <, then 

b 

COS G) = , 

Vac 
and the anharmonic ratio reduces to 

an*-, 
whence we deduce the following theorem : 



206 THE THEORY OF SCREWS. [205- 

The angle between two screws is equal to \i times the logarithm of the an- 
ha,rmonic ratio in which their corresponding chord is divided by the infinite 
pitch conic. 

The reader will be here reminded of the geometry of non-Euclidian 
space, in which a magnitude, which in Chapter XXVI. is called the Intervene, 
analogous to the distance between two points, is equal to ^i times the 
logarithm of the anharmonic ratio in which their chord is divided by the 
absolute. We have only to call the conic of infinite pitch the absolute, and 
the angle between two screws is the intervene between their corresponding 
points. 

206. Screws at Right Angles. 

If two screws, 9 and </>, be at right angles, then 

$]<l + 0. 2 <f).2 + 0;,,(f> 3 = 0. 

In other words, 6 and </> are conjugate points of the conic of infinite pitch, 

V + $+ &- & 

All the screws at right angles to a given screw lie on the polar of the point 
with regard to the conic of infinite pitch. Hence we see that all the screws 
perpendicular to a given screw lie on a cylindroid. This is otherwise obvious, 
for a screw can always be found with an axis parallel to a given direction. 
If, therefore, a cylindroid of the system be taken, a screw of the system 
parallel to the nodal axis of that cylindroid can also be found, and thus we 
have the cylindroid and the screw, which stand in the relation of the pole and 
the polar to the conic of infinite pitch. 

A point on the conic of infinite pitch must represent a screw at right angles 
to itself. Every straight line cuts the conic of infinite pitch in two points, and 
thus every cylindroid has two screws of infinite pitch, and each of these 
screws is at right angles to itself. 

In general, the direction cosines of the nodal axis of a cylindroid are 
proportional to the co-ordinates of the pole of the line corresponding to the 
cylindroid with respect to the conic of infinite pitch. 

207. Reciprocal Screws. 

If lt 0,, 3 be the co-ordinates of a screw, and n </>,, </> 3 those of another 
screw, then it is known, 37, that the condition for these two screws to be 
reciprocal is 



We are thus led to the following theorem, which is of fundamental importance 
in the present investigation : 



208] 



PLANE REPRESENTATION OF THE THIRD ORDER. 



207 



A pair of reciprocal screws are conjugate points with respect to the zero- 
pitch conic. 

From this theorem we can at once draw the following conclusions : 

All the screws of the system reciprocal to a given screw lie upon a 
cylindroid. 

For the locus of points conjugate to 9 is, of course, the polar of 
with respect to the zero-pitch conic, and this polar will correspond to a 
cylindroid. 

On any cylindroid one screw can always be found reciprocal to a given 
screw 9. For this will be the intersection of the polar of 9 with the line 
corresponding to the given cylindroid. 

A triad of co-reciprocal screws will correspond to a self-conjugate triangle 
of the conic of zero-pitch. 

208. The Principal Screws of the System. 

Draw the conic of zero-pitch A, and the conic of infinite pitch B, which 
intersect in the four screws of indeterminate pitch, P,, P 2 , P 3 , P 4 (see fig. 39). 
Draw the diagonals of the complete quadrilateral, and let them intersect in 




Fig. 39. 



the points X, Y, Z. These three points are significant. Take any pair of 
them, X and F; then, by the known properties of conies, X and Y are con 
jugate points with respect to both of the conies A and B. The screws X 



208 THE THEORY OF SCREWS. [208, 



and Fmust therefore be mutually perpendicular and reciprocal. If p 

be the pitches of these screws ; if to be the angle between them, and d their 

perpendicular distance, then the virtual coefficient is one-half of 

(Pe + Pt) cos > ~ do* sin &> ; 

as they are reciprocal, this is zero, and as <o is a right angle, we must have 
d = ; in other words, the screws corresponding to X and Y must intersect 
at right angles. The same may be proved of either of the pairs X and Z or 
Y and Z. The points X, Y, Z must therefore correspond to three screws of 
the system mutually perpendicular, and intersecting at a point. But in the 
whole system there is only a single triad of screws possessing these properties. 
They are the axes in the equations of 201, and are known as the principal 
screws of the system. The three points X, Y, Z being the vertices of a 
self-conjugate triangle with respect to both the conies A and B, and hence to 
the whole system, we have the following theorem : 

The vertices of the conjugate triangle common to the system of pitch conies 
correspond to the three principal screws of the three-system. 

209. Expression for the Pitch. 

Each conic drawn through the four points of indeterminate pitch, P 1} P 2 , 
P 3 , P 4 , is the locus of screws with a given pitch belonging to the system. 
We are thus led to connect the constancy of the pitch at each point of this 
conic with another feature of constancy, viz. that of the anharmonic ratio 
subtended by a variable point of the conic with the four fixed points. The 
connexion between the pitch and the anharmonic ratio will now be 
demonstrated. 

Let 1 , # 2 > 03 be the co-ordinates of any point on the conic, and let a, /3, 7 
be the co-ordinates of one of the four points, say P 1 ; then if 0j, $ 2 , :i be the 
current co-ordinates, the equation of the line joining 9 to P l is 



<f>3 



13, 



=0. 



As we are dealing with the anharmonic ratio of a pencil, we may take any 
section for the calculation of the ratios, and, accordingly, make 

<k = 0; 

and we have for the co-ordinates of the point in which the line joining 
8 and Pj intersects </> :! = 0, the conditions, 



- i 7 



209] PLANE REPRESENTATION OF THE THIRD ORDER. 209 

By changing the sign of a, and then changing the signs of ft aud of 7, we 
shall obtain the four points in which the pencil 0(P 1 , P 2 , P 3 , P 4 ) cuts the 

axis <j> 3 = 0. If four values of j- 1 be represented by k, I, m, n, the required 

02 

anharmomc ratio is, of course, 

(n l)(m k) 
(n-m)(l-ky 

and after a few reductions we find that this becomes 



But we have 

_ 



Eliminating 0^, we find that 2 2 and 0/ disappear also when we make 

tf = p-2-p s , P a = p,-p 1) y 2 = pi-p,, 
and we obtain the following result : 



which gives the following theorem : 

Measure off distances p lf p.,, p s , p, from an arbitrary point on a straight 
line, then the anharmonic ratio of the four points thus obtained is equal to the 
anharmonic ratio subtended by any point of the p-pitch conic at the four points 
of indeterminate pitch. 

It is possible without any sacrifice of generality to make the zero-pitch 
conic a circle. For take three angles A, B, G whose sum is 180 and such 
that the equations 

sin2J. _ sin 2B _ sin 2(7 

Pi Pi p 3 

are satisfied where p l} p 2 , p s are the three principal pitches of the three- 
system. If the fundamental triangle has A, B, C for its angles then the 
equation 

a, 2 sin 2 A + 2 2 sin IB + a/ sin 2(7 = 0, 

is the equation of the zero-pitch conic. It is however a well-known theorem 
in conies that this equation represents a circle with its centre at the ortho- 
centre, that is, the intersection of the perpendiculars from the vertices of the 
triangle on its opposite sides. 

We thus have as the system of pitch conies 

a, 2 sin 24 + * 2 2 sin 2B + a 3 2 sin 2(7 - p (a, 2 + a 2 2 + a, 2 ) = 0. 
B. 14 



210 THE THEORY OF SCREWS. [209 

The centre of a conic of the system has for co-ordinates 

sin A sin B sin G 

sin 2A -p sin 25 -p sin 20 -p 

The locus of the centres of the system of conies is easily seen to pass through 
the vertices of the triangle of reference. It must also pass through the 
orthocentre, and hence by a well-known property it must be an equilateral 
hyperbola. It can also be easily shown that this hyperbola must pass 
through the " symmedian" point of the triangle, i.e. through the centre of 
gravity of three particles at the vertices of the triangle when the mass of each 
particle is proportional to the square of the sine of the corresponding angle. 

In Fig. 40 a system of pitch conies has been shown drawn to scale. The 
sides of the fundamental triangle are represented by the numbers 117, 
189, 244 respectively. The equation to the system of conies, expressed in 
Cartesian co-ordinates for convenience of calculation, is 



= .- 1864116 - + y- ( 2-2135882 - - 6852041 ) 

V p i \ p J 

-1649571^ - 9 46700z - 
25199-54 + * 



P 

Among critical conies of the system we may mention : 

1. The two parabolas for which the pitches are respectively 

8766 and "3089. 

2. The three cases in which the conic breaks up into a pair of straight 
lines for the pitches sin 2 A, sin 25, sin 2(7, respectively. Of these, the first 
alone is a real pair corresponding to the pitch 8256034. The equations of 
these lines are 

x = 7 84y - 978, 



For convenience in laying down the curves the current co-ordinates on 
each conic are expressed by means of an auxiliary angle ; thus, for example, 
in computing points on the hyperbola with the pitch 748984 I used the 
equations 

x = 66 + 26-1 sec 6 + 132 tan 6, 

y = 161-5 + 40-7 sec 0. 

The ellipse with pitch 9 was constructed from the equations 
x = - 367 - 168 cos 351 sin 0, 



209] 



PLANE REPRESENTATION OF THE THIRD ORDER. 



211 




142 



212 THE THEORY OF SCREWS. [ 209, 

The following are the pitches of the several points and curves represented 
Vertices of the Triangle + 992, + 825, - 448, 
Ellipses + -96, + -92, + 1, - 4, 
Parabola + 309, 
Hyperbolas + 748984, + -8300467. 

The locus of the centres of the pitch conies is 
a^j sin (A - B) sin 2(7 + 2 a 3 sin (B - C) sin 1A + a.^ sm(C-A) sin 25 = 0. 

210. Intersecting Screws in a Three-System. 

If two screws of a three-system intersect, then their two corresponding 
points must fulfil some special condition which we propose to investigate. 

Let a be one screw supposed fixed, then we shall investigate the locus of 
the point 6 which expresses a screw which intersects a. We can at once 
foresee a certain character of this locus. A ray through a can only cut it in one 
other point, for if it cut in two points, we should have three co-cylinclroidal 
screws intersecting, which is not generally possible. The locus is, as we shall 
find, a cubic and the necessary condition is secured by the fact that a is a 
double point on the cubic, so that a ray through a has only one more point 
of intersection with the curve. We can indeed prove that this curve must 
be a cubic from the fact that any screw meets a cylindroid in three points. 
Draw then a ray ( 200) corresponding to a cylindroid of the three-system. 
There must be in general three points of the locus on this ray. Therefore the 
locus must be a cubic. 

As a and 6 intersect we have since d 6a = 

2<3- ea = (p a + p e ) cos 0<9). 

By substituting the values of the different quantities in terms of the co 
ordinates we have the following homogeneous equation of the cubic : 

= 2 (0!* + 2 2 + 3 2 ) (p^A +P&A + P&A) (a, 2 + 2 2 + a 3 2 ) 

a 3<?3) (ai 2 + 2 2 + a s 2 ) 



We first note that this cubic must pass through the four points of inter 

section of 

= 0,"+ <9./ + 6*, 

= pA* + p&i + pA 2 - 

But this might have been expected, because as we have shown ( 203) each of 
these four points corresponds, not to a single screw, but to a plane of screws. 



210] PLANE REPRESENTATION OF THE THIRD ORDER. 213 

In a certain sense therefore a must intersect each of these four screws, and 
accordingly the cubic has to pass through the four points. 

To prove that a is a double point we write for brevity 

p = 0^ + e:- + 3 -, R = p&A + p.&A + p 3 a 3 3 , 

Q = Pidi 1 + pA* + P-*0 /, 8 = ^0,+ a,8, + a 3 3 , 
L = p&S + p 2 aj + p s a 3 2 , H = a/ 2 + a, 2 + a 3 2 , 

and the equation is 

P(2HR-LS)-8QH=0. 

Differentiating with respect to 1} 0. 2 , 3 respectively and equating the results 
to zero we have 

= 20j [2RH -LS- aHS] + a, [2aPH -LP- HQ], 
= 20 8 [2RH -LS- bHS] + a 2 [2bPH -LP- HQ], 
= 20 3 [2RH -L8- cHS] + a 3 [2cPH-LP - HQ]. 

These are satisfied by 6 l = a l , 2 = 2 , 3 = a s which proves that a is a double 
point. 

The cubic equation is satisfied by the conditions 



=p ] a l 1 +p 2 a 2 2 +p 3 <x z 3 . 

This might have been expected because these equations mean that o and 
are both reciprocal and rectangular, in which case they must intersect. Thus 
we obtain the following result : 

If !, a 2 , a s are the co-ordinates of a screw a in the plane representation, 
then the co-ordinates of the screw which, together with a, constitute the 
principal screws of a cylindroid of the system are respectively 

Ps-p* Pi- Pi P*-PI 

1 2 3 

The following theorem may also be noted. Among the screws of a three- 
system which intersect one screw of that system there will generally be two 
screws of any given pitch. 

For the cubic which indicates by its points the screws that intersect a 
will cut any pitch conic in general in six points. Four of these are of course 
the four imaginary points referred to already. The two remaining inter 
sections indicate the two screws of the pitch appropriate to the conic which 
intersect a. 

The cubic 

P(2HR-LS)-SQH = Q, 



214 THE THEORY OF SCREWS. [210, 

of course passes through the two points which lie at the intersections of 
2HR LS = and Q = 0. Hence we deduce the following result. 

The cylindroid through the two screws of zero pitch which intersect a 
corresponds to the ray whose equation is 

2( 1 2 +a, 2 +a 3 2 )(> 1 a 1 6> 1 +p,aA + p^-A)-(p^ +p,a.* +p&?} (a^ +a 2 2 +a 3 3 )=0, 



a will of course intersect a third screw on this cylindroid and be perpen 
dicular thereto. 

The cubic passes through the two points defined by 

= aA + oc,<9 2 + a 3 3 , 



but these points are the points of contact of the pair of tangents drawn from 
the point a to the conic of infinite pitch. 

These two points in addition to the four points on the same conic defined 
by its intersection with the conic of zero pitch, make six known points on the 
cubic. If however we are given six points on a cubic and also a double 
point on the curve, then it is determinate. Thus we obtain the following 
geometrical construction. 

In the plane representation of a three-system we draw the conic of zero 
pitch C and the conic of infinite pitch U. To find the locus of the screws 
which intersect a given screw a, it is necessary to draw the two tangents 
from a to If and through the two points of contact, and also the four points 
common to C and U then draw that single cubic which has a double point 
at a. See Appendix, note 5. 

211. Application to Dynamics. 

By the aid of the plane representation we are enabled to solve certain 
problems in the dynamics of a rigid body which has freedom of the third 
order. 

Let an impulsive wrench act upon a quiescent rigid body ; it is required 
to determine the instantaneous screw about which the body will commence to 
twist. 

It has been shown ( 96) that the impulsive wrench, wherever situated, 
can generally be adequately represented by the reduced impulsive wrench 
on a screw of the three-system. The problem is therefore reduced to the 
determination of the point corresponding to the instantaneous screw, when 
that corresponding to the impulsive screw is known. 



211] PLANE REPRESENTATION OF THE THIRD ORDER. 215 

We have first to draw the conic of which the equation is 



This conic is of course imaginary, being in fact the locus of screws about 
which, if the body were twisting with the unit of twist velocity, the kinetic 
energy would nevertheless be zero. If two points 0, (ft are conjugate with 
respect to this conic, then 



The screws corresponding to 6 and $ are then what we have called conjugate, 
screws of inertia. 

This conic is referred to a self-conjugate triangle, the vertices of 
which are three conjugate screws of inertia. There is one triangle self- 
conjugate both to the conic of zero pitch, and to the conic of inertia just 
considered. The vertices of this triangle are of especial interest. Each pair 
of them correspond to a pair of screws which are reciprocal, as well as being 
conjugate screws of inertia. They are therefore what we have designated 
as the principal screws of inertia ( 87). They degenerate into the principal 
axes of the body when the freedom degenerates into the special case of 
rotation around a fixed point. 

When referred to this self-conjugate triangle, the relation between the 
impulsive point and the corresponding instantaneous point can be expressed 
with great simplicity. Thus the impulsive point <, whose co-ordinates are 

#iWi 2 -5- Pi ; 0*uf -j- p 2 ; 3 3 a -r p 3 , 

corresponds to the instantaneous point whose co-ordinates are 6 lt 2 , 3 . The 
geometrical construction is sufficiently obvious when derived from the 
theorem thus stated. 

If <f> denote an impulsive screw, and 6 the corresponding instantaneous screw, 
then the polar of <f> with regard to the conic of zero pitch is the same straight line 
as the polar of 6 with regard to the conic of inertia. 

If H be the virtual coefficient of two screws 6 and 77, then 



It follows that the locus of the points which have a given virtual coefficient 
with a given point is a conic touching the conic of infinite pitch at two 
points. If -v/r be the screw whose polar with regard to the conic of infinite 
pitch is identical with the polar of 77 with regard to the conic of zero pitch, 
then all the screws 6 which have a given virtual coefficient with 77 arc 
equally inclined to ^. It hence follows that all the screws of a three- 
system which have a given virtual coefficient with a given screw are parallel 



216 THE THEORY OF SCREWS. [211 

to the generators of a right circular cone. All the screws reciprocal to 77 
form a cylindroid, and fy is the one screw of the system which is parallel 
to the nodal line of the cylindroid. The virtual coefficient of ^ and 17 is 
greater than that of 77 with any other screw. 

If 6 be a screw about which, when a body is twisting with a given 
twist velocity it has a given kinetic energy, then we must have 

U *e? + uje* + u 3 2 3 2 - E(e? + e.? + e/) = o, 

where E is a constant proportional to the energy. It follows that the locus 
of 6 must be a conic passing through the four points of intersection of the 
two conies 

u*d? + u?e? + u 3 2 6 3 2 = 0, 



The four points in which these two conies intersect correspond to the screws 
about which the body can twist with indefinite kinetic energy. These four 
points A, B, C, D being known, the kinetic energy appropriate to every point 
P can be readily ascertained. It is only necessary to measure the anharmonic 
ratio subtended by P, at A, B, C, D, and to set off on a straight line 
distances w x 2 , u, u 3 2 , h 2 , so that the anharmonic ratio of the four points 
shall be equal to that subtended by P. This will determine h 2 , which is 
proportional to the kinetic energy due to the unit twist velocity about the 
screw corresponding to P. 

A quiescent rigid body of mass M receives an impulsive wrench of given 
intensity on a given screw 77 ; we investigate the locus of the screw 6 belonging 
to the three-system, such that if the body be constrained to twist about 6, 
it shall acquire a given kinetic energy. 

It follows at once (91) that we must have 



where E is proportional to the kinetic energy. The required locus is there 
fore a conic having double contact with the conic of inertia. 

It is easy to prove from this that E will be a maximum if 
V#i : Ptfi = W 2 2 2 : p 2 i] 2 = u 3 2 3 : p s rj 3 ; 

whence again we have Euler s well-known theorem that if the body be 
allowed to select the screw about which it will twist, the kinetic energy 
acquired will be larger than when the body is constrained to a screw other 
than that which it naturally chooses ( 94). 

A somewhat curious result arises when we seek the interpretation of 
a tangent to the conic of infinite pitch. This tangent must, like any other 
straight line, correspond to a cylindroid ; and since it is the polar of the 



211] 



PLANE REPRESENTATION OF THE THIRD ORDER. 



217 



point of contact, it follows that every screw on the cylindroid must be at 
right angles to the direction corresponding to the point of contact. The 
co-ordinates of the point of contact must therefore be proportional to the 
direction cosines of the nodal line of the cylindroid. 

If the body be in equilibrium under the action of a conservative system 
of forces, then there is a conic (analogous to the conic of inertia) which 
denotes the locus of screws about which the body can be displaced to a 
neighbouring position, so that even as far as the second order of small 
quantities no energy is consumed. The vertices of the triangle self-conjugate 
both to this conic and the conic of inertia correspond to the harmonic 
screws about which, if the body be once displaced, it will continue to 
oscillate. 



CHAPTER XVI. 

FREEDOM OF THE FOURTH ORDER. 

212. Screw System of the Fourth Order. 

The most general type of a screw system of the fourth order is exhibited 
by the set of screws which are reciprocal to an arbitrary cylindroid ( 75). 
To obtain certain properties of this screw system it is, therefore, only 
necessary to re-state a few results already obtained. 

All the screws which belong to a screw system of the fourth order and 
which can be drawn through a given point are generators of a certain cone 
of the second degree ( 23). 

All the screws of the same pitch which belong to a screw system of the 
fourth order must intersect two fixed lines, viz. those two screws which, 
lying on the reciprocal cylindroid, have pitches equal in magnitude but 
opposite in sign to the given pitch ( 22). 

One screw of given pitch and belonging to a given screw system of the 
fourth order can be drawn through each point in space ( 123). 

As we have already seen that two screws belonging to a screw system 
of the third order can be found in any plane ( 178), so we might expect 
to find that a singly infinite number of screws belonging to a screw system 
of the fourth order can be found in any plane. We shall now prove that 
all these screws envelope a parabola. A theorem equivalent to this has 
been already proved in a different manner in 162. 

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 system of the 
fourth order can be drawn in a given plane through a given point. But 
it can be easily shown that only one screw of the system parallel to a 
given line can be found in the plane. Therefore from the point at infinity 
only a single finite tangent to the curve can be drawn. Therefore the other 



212, 213] FREEDOM OF THE FOURTH ORDER. 219 

tangent from the point must be the line at infinity itself, and as the line 
at infinity touches the conic, the envelope must be a parabola. 

In general there is one line in each screw system of the fourth order, 
which forms a screw belonging to the screw system, whatever be the pitch 
assigned to it. The line in question is the nodal line of the cylindroid 
reciprocal to the four-system. 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. 

A body which has freedom of the fourth order may be illustrated by the 
particular case where one point P of the body is forbidden to depart from 
a given curve. The position of the body will then be specified by four 
quantities, which may be, for example, 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 has de 
generated to a plane, and in fact consists of screws of zero pitch on all the 
normals to the curve at P. 

It is required to determine the locus of screws parallel to a given straight 
line L, and belonging to a screw system of the fourth order. The problem 
is easily solved from the principle that each screw of the screw system must 
intersect at right angles a screw of the reciprocal cylindroid ( 22). Take, 
therefore, that one screw 6 on the cylindroid which is perpendicular to L. 
Then a plane through 6 parallel to L is the required locus. 

213. Equilibrium with freedom of the Fourth Order. 

When a rigid body has freedom of the fourth order, it is both necessary 
and sufficient for equilibrium, that the forces shall constitute a wrench upon 
a screw of the cylindroid reciprocal to the given screw system. Thus, if a 
single 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 would be impossible for equilibrium to 
subsist under the operation of a single force. It is, however, worthy of 
remark, that if one force could act without disturbing the equilibrium, 
then in general another force (on the other screw of zero pitch) could 
also act without disturbing equilibrium. 

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 equilibrium. In no other case, however, can a body which has freedom 
of the fourth order be in equilibrium under the influence of a couple. 



220 THE THEORY OF SCREWS. [213, 

We can also investigate the conditions under which five forces applied 
to a free rigid body can neutralize each other. The five forces must, as 
the body is free, belong to a screw system of the fourth order. Draw 
the cylindroid reciprocal to the system. The five forces must, therefore, 
intersect both the screws of zero pitch on the cylindroid. We thus prove 
the well-known theorem that if five forces equilibrate two straight lines can 
be drawn which intersect each of the five forces. Four of the forces will 
determine the two transversals, and therefore the fifth force may enjoy any 
liberty consistent with the requirement that it also intersects the same two 
lines. 

If A l ,...A 5 be the five forces, the ratio of any pair, let us say for 
example, A : A z is thus determined. 

Let P, Q be the two screws of zero pitch upon the cylindroid, i.e. the 
two common transversals of A ly ... A 5 . 

Choose any two screws X and Y reciprocal to both A! and J. 2 , but not 
reciprocal to A 3 , A 4 or A 5 . 

Choose any screw Z reciprocal to A 3 , A^, A 5 , but not reciprocal to A l 
or A 2 . 

Construct ( 25) the single screw / reciprocal to the five screws 

X, 7, P, Q, Z. 

The four screws X, Y, P, Q are reciprocal to the cylindroid A 1} A. 2 \ 
therefore /, which is reciprocal to X, Y, P, Q, must lie upon the cylindroid 
(A lt Aj($2Q 

Since P, Q, Z are all reciprocal to A 3 , A t , A 5 , it follows that / being 
reciprocal to P, Q, Z must belong to the screw system A 3 , A t , A 5 . Hence 
/ is found on the cylindroid (A 1} A 3 ), and it must also belong to the system 
(A 3 , AI, A s ). If, therefore, forces along A 1 ,...A 5 are to equilibrate, the 
forces along A 1> A 3 must compound into a wrench on /. 

But / being known by construction the angles A Z I and A^I are known, 
and consequently the ratio of the sines of these angles, i.e. the relative 
intensities of the forces on A 1 and A 2 are determined ( 14). 

If a free rigid body is acted upon by five forces, the preceding con 
siderations will show in what manner the body could be moved so that it 
shall not do work against nor receive energy from any one of the forces. 

Let A!, ... A 5 be the five forces. Draw two transversals L, M intersecting 
A 1 ,...A i . Construct the cylindroid of which L, M are the screws of zero 
pitch ; find, upon this cylindroid, the screw X reciprocal to A 5 . Then the 



214] FREEDOM OF THE FOURTH ORDER. 221 

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 
AU...AS, and therefore a body twisted about X will do no work against 
forces directed along A 1} ... A s . 

From the theory of reciprocal screws it follows that a body rotated 
around any of the lines A ly ... A 5 will not do work against nor receive energy 
from a wrench on X. 

In the particular case, where A l} ... A 5 have a common transversal, then 
X is that transversal, and its pitch is zero. In this case it is sufficiently 
obvious that forces on A^...A 5 cannot disturb the equilibrium of a body 
only free to rotate about X. 

214. Screws of Stationary Pitch. 

We begin by investigating the screws in an ?i- system of which the pitch 
is stationary in the sense employed in the Theory of Maximum and Minimum. 
We take the case of n = 4. 

The co-ordinates O l ,... S of the screws of a four-system have to satisfy 
the two linear equations denning the system. We may write these equations 
in the form 



The screws of reference being co-reciprocal, we have for the pitch p e the 
equation 

SM -.Rp^O, 

where R is the homogeneous function of the second degree in the co 
ordinates which is replaced by unity ( 35) in the formulae after differ 
entiation. 

If the pitch be stationary, then by the ordinary rules of the differential 
calculus ( 38), 



As however belongs to the four-system, the variations of its co-ordinates 
must satisfy the two conditions 



Following the usual process we multiply the first of these equations by 
some indeterminate multiplier \, the second by another quantity p, and then 



222 THE THEORY OF SCREWS. [214 

add the products to the former equation. We can then equate the co 
efficients of 80!,... 8# 6 severally to zero, thus obtaining 



J~D 

2p 6 6 - ja Pe + *A + A^e = 0. 
av 6 

Choose next from the four-system any screw whatever of which the co 
ordinates are $ 15 ... <. Multiply the first of the above six equations by <j> 1} 
the second by < 2 , &c. and add the six products. The coefficients of X and //, 
vanish, and we obtain 

dR , dR 



The coefficient of p g is however merely double the cosine of the angle 
between 6 and <. This is obvious by employing canonical co-reciprocals 
in which 

.R = (0i + 0*y + (0, + e t y + (0, + 0,y, 

whence 

dR dR 



(0, + 2 )+2 (0 3 + 4 ) (0 3 + t ) + (0 8 + 00 (0, + 6 ) = 2 cos 

We thus obtain the following theorem, which must obviously be true for 
other values of n besides four. 

If (f) be any screw of an n-system and if 6 be a screw of stationary pitch 
in the same system then -ar^ = cos (6(f>)p e . 

Suppose that there were two screws of stationary pitch and in an ??- 
system. Then 

13- = cos 



If p e and PCJ, are different these equations require that 

BJ^ = ; cos (#</>) = ; 

i.e. the screws are both reciprocal and rectangular and must therefore 
intersect. 

We have thus shown that if there are two stationary screws of different 
pitches in any w-system, then these screws must intersect at right angles. 

In general we learn that if any screw of an ?i-system has a pitch equal 
to that of a screw of stationary pitch in the same system, then and 
must intersect. For the general condition 

* = cos (#</>) p 6 



214] FREEDOM OF THE FOURTH ORDER. 223 

is of course 

(Pe + P*) cos W) - sin (0<f>) d^ = 2p e cos (0<j>), 



or 

(P<t> ~ Pe) cos (0<f>) - sin (0<f>) d 64> = 0. 

If then p^ = p 6 we must have sin (#(/>) c^ = 0, which requires that and 
must intersect at either a finite or an infinite distance. 

In the case where < is at right angles to 6 it follows from the formula 
^ob cos (0<fi)p 6 that CT04, = 0, or that and are reciprocal. But two screws 
which are at right angles and also reciprocal must intersect, and hence we 
have the following theorem. 

If 6 be a screw of stationary pitch in an n-system, then any other screw 
belonging to the n-system and at right angles to Q must intersect 6. 

If (/> belongs to an w-system its co-ordinates must, on that account, satisfy 
6 n linear equations. If it be further assumed that < has to be perpen 
dicular to 0, then the co-ordinates of < have to satisfy yet one more equa 
tion, i.e. 

. dR , dR 

ft>:-*;BETft 

In this case </> is subjected to 7 n linear equations. It follows ( 76) that 
(f> will have as its locus a certain (n l)-system, whence we have the following 
general theorem. 

If 6 is a screw of stationary pitch in an n-system P then among the 
(n Y)-sy stems included in P there is one Q such that every screw of Q intersects 
at right angles. 

These theorems can also be proved by geometrical considerations. If a 
screw 6 have stationary pitch in an w-system it follows a fortiori that 6 must 
have stationary pitch on any cylindroid through 6 and belonging wholly to 
the n-system. This means that must be one of the two principal screws 
on such a cylindroid. Choose any other screw < of the system and draw the 
cylindroid (0, </>) then 6 is a principal screw, and if 6 and the other principal 
screw on the cylindroid be two of the co-reciprocal screws of reference, then 
the co-ordinate of < with respect to is cos(0<) ( 40). But that co-ordinate 
must also have the general form -nr^ -rp l , whence at once we obtain 



Let be a screw of stationary pitch in a three-system, and let </> and ^r 
be any two other screws in that system. Then 6 is one of the principal 
screws on the cylindroid (#</>) ; let <j be the other principal screw on that 



224 THE THEORY OF SCREWS. [214, 

cylindroid. In like manner let p be the other principal screw of the cylin- 
droid (0i/r). Then p and a determine the cylindroid (per) which belongs to 
the system, 6 must lie on the common perpendicular to p and cr, and hence 
the screws of the cylindroid (per) each intersect 6 at right angles. 

If is a screw of stationary pitch in a four-system, it can be shown 
that three screws p, cr, r not on the same cylindroid can be found in the same 
system, and such that they intersect 6 at right angles. In this case p, cr, r 
will determine a three-system, every screw of which intersects 6 at right 
angles. 

215. Application to the Two-System. 

The principles of the last article afford a simple proof of many funda 
mental propositions in the theory. We take as the first illustration the well- 
known fact ( 76) that if the co-ordinates of a screw satisfy four linear 
equations then the locus of that screw is a cylindroid. 

From the general theorem we see by the case of n = 2 that in any two- 
system a screw of stationary pitch will be intersected at right angles by 
another screw of the two-system. 

These two screws may be conveniently taken as the first and third of the 
canonical co-reciprocal system lying on the axes of x and y. Hence we have 
as the co-ordinates of a screw of the system l , 0, # 3 , 0, 0, 0. 

The investigation has thus assumed a very simple form inasmuch as the 
four linear equations express that of the six co-ordinates of a screw of the 
system four are actually zero. 

Let X, fjb, v be the direction angles of the screw 6 with respect to the 
associated Cartesian axes then ( 44), 

a _ (Pe + a ) cos ^ ~ d ei sin X (p e a) cos X d ei sin X 

V\^ ) v) = I 

a a 

a (Pe + &) cos P - ^02 sin /u. (p e - b) cos p-d^ sin p 

e * = IT " ; ~TF~ - ; 

_ _ ( p e + c) cos v d es sin v _ _ (p 6 c) cos v - d es sin v 
c c 

The two last of these equations give 

cos v = ; d e3 = 0. 

Hence we learn that 6 must intersect the axis of z at right angles. 6 is 
thus parallel to the plane of xy at a distance d ei = d ez = z, and accordingly 
we have the equations 

( p g a) cos X z sin X = 0, 

(p 6 b) cos /j. z sin /u, = 0, 



215] FREEDOM OF THE FOURTH ORDER. 

whence eliminating z and observing that X //, = 90 we obtain, 

p d = a cos 2 X + 6 sin 2 X, 

and eliminating p e , 

(b a) sin X cos X = z. 
If we desire the equation of the surface we have 

y = x tan X, 
and hence finally 



225 



Thus again we arrive at the well-known equation of the cylindroid. 

We can also prove in the following manner the fundamental theorem 
that among the screws belonging to any two-system there are two which 
intersect at right angles ( 13). 

Let 6 be any screw of the two-system, and accordingly the six co-ordinates 
of 9 must satisfy four linear equations which may be written 



If be a screw which intersects 6 at right angles, then we must 
also have 









inasmuch as these screws are reciprocal as well as rectangular. 

From these six equations O l ,...0 6 can be eliminated, and we have the 
resulting equation in the co-ordinates of <f>, 



B. 



dR 


dR 


dR 


dR 


dR 


rf# 


* 


<% 


d(f 3 


* 


** 


d<f> r , 


^Z, 


^t, 


A 3 , 


A, 


A 5 , 


A K 


B,, 


B 2 > 


B 3 , 


% 


B 5} 


B B 


<?i, 


C z , 


G 3 , 


P,, 


c r> , 


c fi 


A, 


A, 


A, 


A, 


A, 


A 



= o. 



15 



226 THE THEORY OF SCREWS. [215- 

This equation involves the co-ordinates of < in the second degree. If this 
equation stood alone it would merely imply that $ belonged to the quadratic 
five-system ( 223) which included all the screws that intersected at right 
angles any one of the screws of the given cylindroid. If we further assume 
that <f> is to be a screw on the given cylindroid, then we have 



... + (7 6 (/> 6 = 0, 



From these five equations two sets of values of <f> can be found. Thus 
among the system of screws which satisfy four linear equations there must 
be two screws which intersect at right angles. These are of course the two 
principal screws of the cylindroid. 

216. Application to the Three-System. 

The equations of the three-system can be also deduced from the principle 
employed in 214 which enunciated for this purpose is as follows. 

If 6 be a screw of stationary pitch in a three-system P then there is a 
cylindroid belonging to P such that every screw of the cylindroid intersects 
6 at right angles. 

It is obvious that this condition could only be complied with if lies on 
the axis of the cylindroid, and as the cylindroid has two intersecting screws 
at right angles we have thus a proof that in any three-system there must be 
one set of three screws which intersect rectangularly. Let their pitches be 
a, b, c, then on the first we may put a screw of pitch a, on the second a 
screw of pitch b, and on the third a screw of pitch c. Thus we arrive 
at a set of canonical co-reciprocals specially convenient for the particular 
three-system. 

We have therefore learned that whatever be the three linear equations 
defining the three-system it is always possible without loss of generality to 
employ a set of canonical co-reciprocals such that the 1st, 3rd and 5th screws 
shall belong to the system. 

These three screws will define the system. Any other screw of the 
system can be produced by twists about these three screws. Hence we 
see that for every screw of the system we must have 

0, = ; 4 = 0; 6 = 0. 



217] FREEDOM OF THE FOURTH ORDER. 227 

If X, /j,, v be the direction angles of 6 we have therefore ( 44) 

fi _ (p e a) cos X d ei sin X _ 
c/ 2 (j 

a 

a (Pd - b) cos /u, - d e3 sin /i 
4 - -g- :0, 

/, _ (P - c) cos v - d 65 sin v 

t/G = U. 



c 
The direction cosines of the common perpendicular to 6 and 1 arc 

cos v cos //, 

sin X sin X 

whence the cosine of the angle between this perpendicular and the radius 
vector to a point ac, y, z on is 

y cos v z cos /j, d 6l 
r sin X r sin X r 

or d ei sin X = y cos v z cos /x. 

We have thus the three conditions 

(p e a)cos\ +z cos //, ?/ cos z/ = (i), 

2 cos X + (p 9 b) cos /i + cos v = (ii), 

+ ycosX # cos //, -f ( p Q c) cos v = (iii), 

whence eliminating cos X, cos //,, cos v we obtain 

( p e a) (p e b) (p e c) + (p 6 a)x 1 -\-(p e H)y- + (p e c)z" = 0. 

Thus we deduce the equation otherwise obtained in 174, for the family of 
pitch-hyperboloids on which are arranged according to their pitches the 
several screws of the three-system. 

217. Principal pitches of the Reciprocal Cylindroid. 

From a system of the fourth order a system of canonical co-reciprocals 
can in general be selected which possesses exceptional facilities for the investi 
gation of the properties of the screws which form that four-system. 

Let OA and OB be the axes of the two principal screws of the reciprocal 
cylindroid. Let a and b be the pitches of these two principal screws and 
let c be any third linear magnitude. Let 00 be the axis of the cylindroid. 

Then the canonical co-reciprocal system now under consideration consists of 
Two screws on OA with pitches + a and - a. 
Two screws on OB with pitches + b and - b. 
Two screws on 00 with pitches + c and c. 

152 



228 THE THEORY OF SCREWS. [217- 

Of these the four screws with pitches a,b,+c,-c respectively are each 
reciprocal to the cylindroid. Each of these four screws must thus belong to 
the four-system. Further these four screws are co-reciprocal. 

If 0^ ... ({ be the six co-ordinates of a screw in the four-system referred 
to these canonical co-reciprocals, then we have 

l = 0, 3 = 0. 
For 0! = ie , but as the first screw of reference belongs to the reciprocal 

CL 

cylindroid we must have tzr ]0 = 0. In like manner vr 3g = 0, and therefore 
0! = and #3 = are the two linear equations which specify this particular 
four-system. 

The pitch of any screw on the four-system expressed in terms of its 
co-ordinates is 

I^02^j>04 2 + c (0 5 2 - 6 2 ) 
*f+0t + & + lff 

of which the four stationary values are a, b, + oo , GO . 

We may remark that if the four co-ordinates here employed be taken as 
a system of quadriplanar co-ordinates of a point we have a representation 
of the four- system by the points in space. P^ach point corresponds to one 
screw of the system. The screws of given pitch p e are found on the quadric 

surfaces 

U+p e V=0, 

where ?7=0 is the quadric whose points correspond to the screws of zero 
pitch and where V= is an imaginary cone whose points correspond to the 
screws of infinite pitch. Conjugate points with respect to U = will cor 
respond to reciprocal screws. A plane will correspond to a three-system 
and a straight line to a two-system. 

The general theorem proved in 214 states that when is a screw of 
stationary pitch in an ra-system to which any other screw < belongs, then 

^e* = Pe cos 0(/>. 

Let us now take a four-system referred to any four co -reciprocals and choose 

for $ in the above formula each one of the four co-reciprocals in succession, 

we then have 

Pi e, =p e {0, + 2 cos (12) + 6 3 cos (1.3) + 4 cos (14)}] 

p 2 2 = p e {6 l cos (1 2) + 2 +0 3 cos (23) + 4 cos (24)} 

p 3 3 = Pe {0i cos (1.3) + 2 cos (23) + a + 4 cos (34)} 

2>404 = p s {0i cos (14) + 2 cos (24) + 6, cos (34) + 4 ] 

Eliminating 1( 2 , 3) 4 we deduce a biquadratic for p e . But we have 



219] FREEDOM OF THE FOURTH OKDEK. 229 

already seen that two of the roots of this must be infinite, whence this 
equation reduces to a quadratic, and its roots are as we have seen equal 
but opposite in sign to the pitches of the principal screws of the reciprocal 
cylindroid. 

After a few reductions and replacing p e by p we obtain the following 
equation 

P*(P!P* sin 2 (34) + p,p a sin 2 (24) + . . .) 



We thus deduce from any four co-reciprocal screws the quadratic equation 
which gives the pitches of the two principal screws of the cylindroid to 
which the given four-system is reciprocal. 

218. Equations to the screw in a four-system </ 

The screws of the four-system are defined by the equations 

(p e + a) cos X + z cos yu, y cos v = 0, 
z cos X + ( p e + b) cos fj, + x cos v = 0, 

where p g is the pitch where cos X, cos p, cos v are the direction cosines and 
where oc, y, z is a point on the screw. By these equations the properties of 
the various screws of the system can be easily investigated. 

If p e be eliminated we obtain 

x cos X cos v + y cos p cos v z (cos 2 X + cos 2 p,) (a 6) cos X cos //., 

whence we obtain for the equation to the cone of screws which belongs to the 
four-system, and has its vertex at #, y 0) Z Q 

x (as - x,} (z - z,} + 7/0 (y - y ) (z - z,) 

- z, {(x - a? ) 3 + (y - y ) 2 } ~(a-b) (x - x ) (y - // ) = 0. 
This is of course the cone which has been referred to in 123. 

219. Impulsive Screws and Instantaneous Screws. 

A body which is free to twist about all the screws of a screw system of 
the fourth order receives an impulsive wrench on the screw 77, the impulsive 
intensity being 77 ". It is required to calculate the co-ordinates of the screw 
6 about which the body will commence to twist, and also the initial re 
actions of the constraints. 

Let X and /j, be any two screws on the reciprocal cylindroid, then the 
impulsive reaction of the constraints may be considered to consist of 
impulsive wrenches on X, p of respective intensities X ", //," . If we adopt 



230 THE THEORY OF SCREWS. [219- 

the six absolute principal screws of inertia as screws of reference, ( 79) 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 proportional to p^, ..., p 6 6 6 . 
It follows that the given impulsive 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 certain quantity which 
is the same for each co-ordinate, we have the six equations 



Multiply the first of these equations by X 1; the second by Xj, &c.: adding 
the six equations thus obtained, and observing that 6 is reciprocal to X, and 
that consequently 

2M\i = 0, 
we obtain 

f} / " ^r} 1 \ 1 + X "^! 2 + //"SXj//,! = 0, 

and similarly multiplying the original equations by p 1 , ..., ^ 6 and adding, 
we obtain 

^x, + /"S 2 = 0. 



From these two equations the unknown quantities X ", //" can be found, 
and thus the initial reaction of the constraints is known. Substituting the 
values of X ", //" in the six original equations, the co-ordinates of the 
required screw 9 are determined. 

220. Principal Screws of Inertia in the Four-System. 

We have already given in Chapter VII. the general methods of deter 
mining the principal screws of inertia in an ?i-system. The following is a 
different process which though of general application is in this chapter set 
down for the case of the four-system. 

Choose four co-reciprocal screws a, ft, 7, 8 of the four-system and let 
their co-ordinates be as usual !,... ,fc; &,...,&; y n ...,7; 81, ...,8 6 ; 
referred to the six absolute principal screws of inertia ( 79). 

Let an impulsive wrench on one of the principal screws of inertia 6 in 
the four-system be decomposed into components on a, /3, 7, 8, and let the 
impulsive intensities be a ", ", 7 ", 8 ". 

Let X, p, be any two screws on the reciprocal cylindroid. Then the body 
will move as if it had been free and had received impulsive wrenches on the 
absolute principal screws of inertia, the impulsive intensities being 



a" a 6 + {3" j3 G + y " 7ti + B "S V + X /7/ X 6 



221] FREEDOM OF THE FOURTH ORDER. 231 

The co-ordinates of are proportional to 

/// , rt/rr r\ . ill , 5/"S> 

a a x + p & + 7 71 + o di 



/// . nni rt i i 

a a + /3 #, + 7 7e + <% 

As # is to be a principal screw of inertia it follows that the expressions 
last written multiplied severally by p l} ...,p 6 must be proportional to the 
intensities of the impulsive wrenches received by the body : whence we have 
the following equations in which h is a quantity which is the same for each 
of the co-ordinates. 

i + 7" 7i + 8 "^ = *" i + P"& + 7"Vi + S/// 



We are now to multiply these equations by a u . .., a fi respectively, and 
add. If we repeat the process using p\, ...,/3 6 ; 71, ..., 7el !,..., S 6 ; 
X ls ...,X 6 ; fr, ...,fj, 6 and if we remember that a is reciprocal to /3, 7, 8 
because the system is co-reciprocal and that a is reciprocal to A, and p, 
because X and //, belong to the reciprocal system, then observing that like 
conditions hold for ft, 7, and S, we have the equations 



// S7 1 X 1 +/ // S7,/* 1 =0, 
/ S8 1 X 1 + /i // SV 1 =0, 
V +/ // S\ 1 ^ 1 =0, 



From these equations a ", /S" , 7 ", 8 ", X" , /*" can be eliminated and the 
result is to give a biquadratic for h. Thus we have the four roots for the 
equation. Each of these roots will give a corresponding set of values for 
" , ", 7 ", 8 ", X ", p" thus we obtain 



which are proportional to the co-ordinates of the corresponding principal 
screw of inertia. 

The values of X " and ///" determine the impulsive reaction of the con 
straints. 

221. Application of Euler s Theorem. 

It may be of interest to show how the co-ordinates of the instantaneous 



232 THE THEORY OF SCREWS. [221- 

screw corresponding to those of a given impulsive screw can be deduced 
from Euler s theorem ( 94). If a body receive an impulsive wrench on a 
screw 7] while the body is constrained to twist about a screw 6, then we have 
seen in 91 that the kinetic energy acquired is proportional to 



If #j, 0. 2 , 3 , #4 be the co-ordinates of referred to the four principal 
screws of inertia belonging to the screw system of the fourth order, then 

(195,97) 



uf = uf + w;- <, 2 + u./* + ufOf. 

Hence we have to determine the four independent variables 1} 6.,, 3) # 4 , so 
that 



shall be stationary. This is easily seen to be the case when lt 0.,, 3 , 4 are 
respectively proportional to 

Jlfci *, |9t, ~^. 

U/l U 2 ^3 Ul 

These are accordingly, as we already know ( 97), the co-ordinates of the 
screw about which the body will commence to twist after it has received 
an impulsive wrench on ?/. 

This method might of course be applied to any order of freedom. 

222. General Remarks. 

It has been shown in 80 how the co-ordinates of the instantaneous 
screw corresponding to a given impulsive screw can be determined when the 
rigid body is perfectly free. It will be observed that the connexion between 
the two screws depends only upon the three principal axes through the 
centre of inertia, and the radii of gyration about these axes. We may 
express this result more compactly by the well-known conception of the 
momental ellipsoid. The centre of the momental ellipsoid is at the centre 
of inertia of the rigid body, the directions of the principal axes of the 
ellipsoid are the same as the principal axes of inertia, and the lengths of 
the axes of the ellipsoid are inversely proportional to the corresponding 
radii of gyration. When, therefore, the impulsive screw is given, the 
momental ellipsoid alone must be capable of determining the corresponding 
instantaneous screw. 

A family of rigid bodies may be conceived which have a common 



FREEDOM OF THE FOURTH ORDER. 233 

momental ellipsoid; every rigid body which fulfils nine conditions will 
belong to this family. If an impulsive wrench applied to a member of 
this family cause it to twist about a screw 0, then the same impulsive 
wrench applied to any other member of the same family will cause it 
likewise to twist about 0. If we added the further condition that the masses 
of all the members of the family were equal, then it would be found that 
the twist velocity, and the kinetic energy acquired in consequence of a 
given impulse, would be the same to whatever member of the family the 
impulse were applied ( 90, 91). 

223. Quadratic n-systems. 

We have always understood by a screw system of the nth order or briefly 
an w-system, the collection of screws whose co-ordinates satisfy a certain 
system of 6 n linear homogeneous equations. We have now to introduce 
the conception of a screw system of the nth order and second degree or briefly 
a quadratic n-system (n < 6). By this expression we are to understand a 
collection of screws such that their co-ordinates satisfy 6 n homogeneous 
equations ; of these equations 5 n, that is to say, all but one are linear ; 
the remaining equation involves the co-ordinates in the second degree. 

Let #!,...,#<; be the co-ordinates of a screw belonging to a quadratic 
w-system. We may suppose without any loss of generality that the 5 n 
linear equations have been transformed into 

Q n+2 = ; n +3 = ; . . . K = 0. 

The remaining equation of the second degree is accordingly obtained by 
equating to zero a homogeneous quadratic function of 

A A 

t/j ... t7 7 l+l 

We express this equation which characterizes the quadratic ?i-system as 



All the screws whose co-ordinates satisfy the 5 n linear equations must 
themselves form a screw system of the 6 (5 n) = (n + l)th system. This 
screw system may be regarded as an enclosing system from which the screws 
are to be selected which further satisfy the equation of the second degree 
[70 = 0. The enclosing system comprises the screws which can be formed by 
giving all possible values to the co-ordinates 6 l , ...,0 n+l . 

Of course there may be as many different screw systems of the nth order 
and second degree comprised within the same enclosing system as there can 
be different quadratic forms obtained by annexing coefficients to the several 
squares and products of n + 1 co-ordinates. If n = 5, the enclosing system 
would consist of every screw in space. 



234 THE THEORY OF SCREWS. [224, 

224. Properties of a Quadratic Two-system. 

The quadratic two-system is constituted of screws whose coordinates satisfy 
three linear equations and one quadratic equation, and these screws lie 
generally on a surface of the sixth degree ( 225). If we take the plane 
representation of the three-system given in Chapter XV., then any conic in 
the plane corresponds to a quadratic two-system and all the points in the 
plane correspond to the enclosing three-system. Since any straight line in 
the plane corresponds to a cylindroid in the enclosing system and the 
straight line will, in general, cut a conic in the plane in two points, we have 
the following theorem. 

A quadratic two-system has two screws in common with any cylindroid 
belonging to the enclosing three-system. 

A pencil of four rays in the plane will correspond to four cylindroids 
with a common screw, which we may term a pencil of cylindroids. Any 
fifth transversal cylindroid belonging also to the same three-system will be 
intersected by a pencil of four cylindroids in four screws, which have the 
same anharmonic ratio whatever be the cylindroid of the three-system 
which is regarded as the transversal. We thus infer from the well-known 
anharmonic property of conies the following theorem relative to the screws 
of a quadratic two-system. 

If four screws a, /3, j, 8 be taken on a quadratic two-system, and also 
any fifth screw 77 belonging to the same system, then the pencil of cylindroids 
(770), (77/3), (^7), (?)&) will have the same anharmonic ratio whatever be the 
screw V). (See Appendix, note G.) 

The plane illustration will also suggest the instructive theory of Polar 
screws which will presently be stated more generally. Let 7=0 be the conic 
representing the quadratic two-system and let V= be the conic representing 
the screws of zero pitch belonging to the enclosing three-system. Let P be a 
point in the plane corresponding to an arbitrary screw 6 of the three-system. 
Draw the polar of P with respect to U=Q and let Q be the pole of this 
straight line Avith respect to V= 0, then Q will correspond to some screw </> of 
the enclosing three-system. From any given screw 6, then by the help of the 
quadratic two-system a corresponding screw $ is determined. We may term 
</> the polar screw of with respect to U0. Three screws of the enclosing 
system will coincide with their polars. These will be the vertices of the 
triangle which is self-conjugate with respect both to U and to V. 

A possible difficulty may be here anticipated. The equation V= is itself 
of course equivalent to a certain quadratic two-system and therefore should 
correspond to a surface of the sixth degree. We know however ( 173) that 
the locus of the screws of zero pitch in a three-system is an hyperboloid, so 
that in this case the expectation that the surface would rise to the sixth 



225J FREEDOM OF THE FOURTH ORDER. 235 

degree seems not to be justified. It is however shown in 202 that this 
hyperboloid is really not more than a part of the locus. There are also four 
imaginary planes which with the hyperboloid complete the locus, and the 
combination thus rises to the sixth degree. 

225. The Quadratic Systems of Higher Orders. 

If we had taken n = 3, then of course the quadratic three-system would 
mean the collection of screws whose four co-ordinates satisfied an equation 
which in form resembles that of a quadric surface in quadriplanar co 
ordinates. A definite number of screws belonging to the quadratic three- 
system can in general be drawn through every point in space. 

We shall first prove that the number of those screws is six. Let l} ..., 8 
be the co-ordinates of any screw referred to a canonical co-reciprocal system. 
Then if x , y , z be a point on 0, we have ( 43) 

(0 5 + 0.) y - (0 3 + 4 ) / = a (6, - 0,) - p e (e l + 0,), 

(0, + 0,) z 1 - (0 3 + (i ) x = & (0 3 - 4 ) - Pe (03 + 0.1 

(0 3 + 04) x - (0, + 0,) y = c(0 6 - 6 ) - p e (0 5 + 6 ). 

If we express that belongs to the enclosing four-system we shall have two 
linear equations to be also satisfied by the co-ordinates of 0. These equations 
may be written without loss of generality in the form 

2 = ; 4 = 0. 

We have finally the equation U g = characteristic of the quadratic three- 
system. From these equations the co-ordinates are to be eliminated. But 
the eliminant of k equations in (k l) independent variables is a homo 
geneous function of the coefficients of each equation whose order is, in 
general, equal to the product of the degrees of all the remaining equations*. 
In the present case, the coefficient of each of the first three equations must 
be of the second degree in the eliminant and hence, the resulting equation 
for p e is of the sixth degree, so that we have the following theorem. 

Of the screws which belong to a quadratic three-system, six can be drawn 
through any point. 

As the enclosing system in this case is of the fourth order, the screws of 
the enclosing system drawn through any point must lie on a cone of the 
second degree ( 218). Hence it follows that the six screws just referred to 
must all lie on the surface of a cone of the second degree. 

We may verify the theorem just proved by the consideration that if the 
function U 6 could be decomposed into two linear factors, each of those factors 

* Salmon, Modern Higher Algebra, p. 76, 4th Edition (1885). 



236 THE THEORY OF SCREWS. [225 

equated to zero would correspond to a three-system selected from the 
enclosing four-system. We know (| 176) that three screws of a three-system 
can be drawn through each point. We have, consequently, three screws 
through the point for each of the two factors of U e , i.e. six screws in all. 

The equation of the 6th degree in p e contains also the co-ordinates x , y , z 
in the sixth degree. Taking these as the current co-ordinates we may 
regard this equation as expressing the family of surfaces which, taken 
together, contain all the screws of the quadratic three-system. The screws 
of this system which have the same pitch p e are thus seen to be ranged on 
the generators of a ruled surface of the sixth degree. All these screws 
belong of course to the enclosing four-system, and as they have the 
same pitches, they must all intersect the same pair of screws on the 
reciprocal cylindroid ( 212). It follows that each of these pitch surfaces of 
the sixth degree must have inscribed upon it a pair of generators of the 
reciprocal cylindroid. 

Ascending one step higher in the order of the enclosing system we see 
that the quadratic four-system is composed of those screws whose co-ordinates 
satisfy one linear homogeneous equation L = 0, and one homogeneous 
equation of the second degree U Q. We may study these screws as 
follows. 

Let the direction cosines of a screw 6 be cos X, cos /u,, cos v. If the 
reference be made, as usual, to a set of canonical co-reciprocals we have 

cos X = B! + 6., ; cos //, = # :i + 64 ; cos v = 5 + # 6 . 

We therefore have for a point x , y 1 , z on 6 the equations ( 218) 
2a0! = (a + p e } cos A, z cos JJL + y cos v , 
2a0 2 = (a p e ) cos X + z cos /* y cos v, 

with similar expressions for 3 , # 4 , 5 , 6 fi . 

Substituting these expressions in L = and U = and eliminating p e , 
we obtain an homogeneous equation of the fourth degree in cos X, 
cos fi, cos v. If we substitute for these quantities x x, y y , z z , we 
obtain the equation of the cone of screws which can be drawn through 
x , y , z \ this cone is accordingly of the fourth degree. We verify this con 
clusion by noticing that if U = were the product of two linear functions, 
this cone would decompose into two cones of the second degree, as should 
clearly be the case ( 218). 

It remains to consider the Quadratic Five-system. In this case the 
enclosing system includes every screw in space, and the six co-ordinates of 



225] FREEDOM OF THE FOURTH ORDER. 237 

the screw 6 are subjected to no other relation than that implied by the 

quadratic relation 

U e = Q. 

As before we may substitute for lt ..., 6 6 from the equations 
2a#j = (a 4- p e ) cos \ z cos fj> + y cos v, 
2a# 2 = (a Pe) cos A. + 2 cos i*> y cos v, 
with similar expressions for 26^, 260 2 > &c. 

Introducing these values into 

U e = 0, 

we obtain a result which may be written in the form 



where A, B, C contain eosX, cos/i, cos v in the second degree, and where 
x , y , z enter linearly into B and in the second degree into G. 

Hence we see that on any straight line in space there will be in general 
two screws belonging to any quadratic five-system. For the straight line 
being given x , y , z are given, and so are cos X, cos //., cos v. The equation 
just written gives two values for a pitch which will comply with the 
necessary conditions. 

If we consider p e and also x , y , z as given, and if we substitute for 
cos X, cos /j,, cos v the expressions x af y y y , z z respectively, we obtain 
the equation of a cone of the second degree. Thus we learn that for each 
given pitch any point in space may be the vertex of a cone of the second 
degree such that the generators of the cone when they have received the 
given pitch are screws belonging to a given quadratic five-system. 

If the equation 



be satisfied, then the straight lines which satisfy this condition will be 
singular, inasmuch as each contains but a single screw belonging to the 
quadratic five-system. As cos \, cos /j,, cos v enter to the fourth degree into 
this equation it appears that each point in space is the vertex of a cone of 
the fourth degree, the generators of which when proper pitches are assigned 
to them will be singular screws of the quadratic five-system. 

If we regard cos X, cos p, cos v as given quantities in the equation 



then this will represent a quadric surface inasmuch as x , y , z enter to the 
second degree. This quadric is the locus of those singular screws of the 
quadratic five-system which are parallel to a given direction. Hence the 
equation must represent a cylinder. 



238 THE THEORY OF SCREWS. [225, 

If B = the two roots of the equation in p e will be equal, but with 
opposite signs; as cos X, cos/i, cosy enter to the second degree in B it 
follows that through any point in space as vertex a cone of the second degree 
can be drawn such that each generator of this cone when the proper pitch is 
assigned to it will equally belong to the quadratic five-system, whether that 
pitch be positive or negative. 

If B = and (7 = 0, then both values of p e must be zero. Regarding 
as , y , z as fixed, each of these equations will correspond to a cone with 
vertex at x , y , z \ these cones will have four common generators, and hence 
we see that through any point in space four straight lines can in general be 
drawn such that with the pitch zero but not with any other pitch, these 
screws will be members of a given quadratic five- system. 

226. Polar Screws. 

The general discussion of the quadratic screw-systems is a subject of 
interest both geometrical and physical. We shall here be content with a 
few propositions which are of fundamental importance. 

Let as before 

U e = Q 

be the homogeneous relation between the co-ordinates 1 ,...,0 n+1 of the 
screws which constitute a quadratic ?i-system. 

Let 77 and denote any two screws other than 6 and chosen from the 
enclosing w-system, from which the screws of the quadratic n-system are 
selected by the aid of the condition U 6 = 0. If then we adopt the fertile 
method of investigation introduced by Joachimsthal, we shall substitute in 
mU e = for 6 1} ..., 6 n+1 the respective values 



The result will be 



TT <* r, 

where U* = -^- + ... + l+1 -- . 

C"7i ar) n+l 

Solving this quadratic equation for I -f- m we obtain two values of this 
ratio and hence ( 119) we deduce the following theorem. 

Any cylindroid of a given (n + l)-system will possess generally two screws 
belonging to every quadratic n-system which the given (n + \)-system 
encloses. 

If the two screws 77 and had been so selected that they satisfied the 
condition 



226] FREEDOM OF THE FOURTH ORDER. 239 

then the two roots of the quadratic are equal but with opposite signs, and 
hence ( 119) we have the following theorem. 

If the condition U^ = is satisfied by the co-ordinates of two screws 77 
and which belong to the enclosing (n + l)-system, then these two screws 77, 
and the tw.o screws which, lying on the cylindroid (rj, ), also belong to the 
quadratic ?i-system U 9 = Q, will be parallel to the four rays of an harmonic 
pencil. 

We are now to develop the conception of polar screws alluded to in 224, 
and this may be most conveniently done by generalizing from a well-known 
principle in geometry. 

Let be a point and S a quadric surface. Let any straight line through 
cut the quadric in the two points X 1 and X 2 . Take on this straight line 
a point P so that the section OXP^X^ is harmonic ; then for the different 
straight lines through the locus of P is a plane. This plane is of course 
the well-known polar of P. We have an analogous conception in the present 
theory which appears as follows. 

Take any screw 77 in the enclosing (n + l)-system. Draw a pencil of n 
cylindroids through 77, all the screws of each cylindroid lying in the enclosing 
(n+ l)-system. Each of these cylindroids will have on it two screws which 
belong to the quadratic w-systetn U e = 0. On each of these cylindroids a 
screw can be taken which is the harmonic conjugate with respect to 77 
with reference to the two screws of the quadratic n-system which are found 
on the cylindroid. We thus have n screws of the f type, and these u screws 
will define an n-system which is of course included within the enclosing 
(n + l)-system. 

The equation of this n-system is obviously 



This equation is analogous to the polar of a point with regard to a 
quadric surface. We have here within a given enclosing (n + l)-system a 
certain re-system which is the polar of a screw 77 with respect to a certain 
quadratic n-system. 

The conception of reciprocal screws enables us to take a further im 
portant step which has no counterpart in the ordinary theory of poles and 
polars. The linear equation for the co-ordinates of f, namely 

tf* = 0, 

is merely the analytical expression of the fact that f is reciprocal to the 



240 THE THEORY OF SCREWS. [226, 

screw of the enclosing (n + l)-system whose co-ordinates are proportional 

to 

1 



P\ rj p n+1 rj n+1 

This we shall term the polar screw of rj with respect to the quadratic 
w-system. It is supposed, of course, that the screws of reference are co- 
reciprocals. 

If CL and ft be two screws of an enclosing (n + 1 )-system, and if i) and 
be their respective polar screws with reference to a quadratic n-system, then 
when a is reciprocal to % we shall have /3 reciprocal to 77. For we have, where 
h is a common factor, 

1 dU a 1 dU a 

htji = -- r~ > ir )n+i i j 

P! da, p n+l da n+l 

whence 

h(p 1 1J 1 /3 1 + ... + Pn+i Vn+i Pn+l) = Uop- 

If therefore $ and rj are reciprocal the left-hand member of this equation is 
zero and so must the right-hand member be zero. But the symmetry shows 
that and a are in this case also reciprocal. We may in such a case regard 
a and ft as two conjugate screws of the quadratic ?i-system. 

As a first illustration of the relation between a screw and its polar, we 
shall take for U a = 0, the form 

Pitf + p*<tf + . . . + p 6 a<? - p (i 2 + 2 2 + . . . + 6 - + 2 1 2 cos (12) . . . ) = 0. 
This means of course that U a = denotes every screw which has the pitch p. 

Take any screw a and draw a cylindroid through a. The two screws of 
pitch p on this cylindroid belong to U and a fourth screw 9 may be taken 
on this cylindroid so that a, 6, and the two screws of pitch p form an 
harmonic pencil. 

By drawing another cylindroid through a. another screw of the 0-system 
can be similarly constructed. If these five cylindroids be drawn through 
we can construct five different screws of the ^-system. To these one screw 
will be reciprocal, and this is the polar of a. We have thus the means of 
constructing the polar of a. 

Seeing however that U a = includes nothing more or less than all the 
p-pitch screws in the universe and that in the construction just given for 
the polar of a there has been no reference to the screws of reference, sym 
metry requires that the polar of a must be a screw which though different 
from a must be symmetrically placed with reference thereto. The only 
method of securing this is for the polar of a with respect to this particular 
function to lie on the same straight line as a. 



227] FREEDOM OF THE FOURTH ORDER. 241 

Hence we deduce that the screw with co-ordinates 

i, Oj) 6> 

and the screw with co-ordinates proportional to 

.1 d JL 1 ^ 1 ^ 

p l d^ p 2 da z p 6 dot G 
in which U is the expression 

P^ + p 2 2 2 ... + p 6 a<? + X (i 2 + 02 2 ... + 2 ai a 2 cos (12) ...) 
must be collinear, and this is true for all values of X. 

We hence see that the co-ordinates of a screw collinear with a must be 
proportional to 



where 

E = a 1 2 + a 2 2 +... + 2a 1 a 2 cos (12) + ... 

Thus we obtain the results of 47 in a different manner. 

227. Dynamical application of Polar Screws. 

We have seen ( 97) that the kinetic energy of a body twisting about a 

7/n/ 

screw 6 with a twist velocity -^- and belonging to a w-system is 



w 



the screws of reference being the principal screws of inertia. 

If we make i< 1 2 1 2 + ... + u n 2 n 2 = 0, then 6 must belong to a quadratic 
-system. This system is, of course, imaginary, for the kinetic energy of 
the body when twisting about any screw which belongs to it is zero*. 

The polar 77 of the screw 0, with respect to this quadratic w-system, has 
co-ordinates proportional to 

u i n u n / 

U 1} ... p. 

Pi Pn 

Comparing this with 97, we deduce the following important theorem : 

A quiescent rigid body is free to twist about all the screws of an enclosing 
(n + \}-system A. If the body receive an impulsive wrench on a screw 17 

* In a letter to the writer, Professor Klein pointed out many years ago the importance of the 
above screw system. He was led to it by expressing the condition that the impulsive screw 
should be reciprocal to the corresponding instantaneous screw. 

B. 16 



242 THE THEORY OF SCREWS. [227, 

belonging to A, then the body will commence to twist about the screw 6, of 
which V) is the polar with respect to the quadratic n-system composed of 
the imaginary screws about which the body would twist with zero kinetic 
energy. 

If a rigid body which has freedom of the nth order be displaced from a 
position of stable equilibrium under the action of a system of forces by a 
twist of given amplitude about a screw 6, of which the co-ordinates referred 
to the n principal screws of the potential are O l ,... n , then the potential 
energy of the new position may, as we have seen ( 103) be expressed by 



If this expression be equated to zero, it denotes a quadratic rc-system, 
which is of course imaginary. We may term it the potential quadratic 

w-system. 

The potential quadratic w-system possesses a physical importance in 
every respect analogous to that of the kinetic quadratic n-system : by 
reference to ( 102) the following theorem can be deduced. 

If a rigid body be displaced from a position of stable equilibrium by a twist 
about a screw 6, then a wrench acts upon the body in its new position on 
a screw which is the polar of 6 with respect to the potential quadratic 
w-system. 

The constructions by which the harmonic screws were determined in the 
case of the second and the third orders have no analogies in the fourth order. 
We shall, therefore, here state a general algebraical method by which they 
can be determined. 

Let U=0 be the kinetic quadratic ?i-system, and F=0 the potential 
quadratic w-system, then it follows from a well-known algebraical theorem 
that one set of screws of reference can in general be found which will reduce 
both U and V to the sum of n squares. These screws of reference are the 
harmonic screws. 

We may here also make the remark, that any quadratic w-system can 
generally be transformed in one way to the sum of n square terms with 
co-reciprocal screws of reference; for if U and p e be transformed so 
that each consists of the sum of n square terms, then the form for the 
expression of p e ( 38) shows that the screws are co-reciprocal. 

228. On the degrees of certain surfaces. 

We have already had occasion ( 210) to demonstrate that the general 
condition that two screws shall intersect involves the co-ordinates of each 



22 8 J FREEDOM OF THE FOURTH ORDER. 243 

of the screws in the third degree. We can express this condition as a 
determinant by employing a canonical system of co-reciprocals. For if two 
screws 6 and (f> intersect, then there must be some point x, y, z which shall 
satisfy the six equations ( 43) : 

( 5 + ) ?/ - ( 3 + 4 ) z = a (! - 03) -p a (! + 2 ), 
(! + a 2 ) z - ( 5 + 6 ) x = b ( 3 - 4 ) - p a (a s 4- 04), 
(a s + a 4 ) x - (! + 03) y = c ( 5 - 6 ) -p a (o s + a,,), 
(0 B + 6 ) y - (0, + 0<) z = a (6, - 2 ) -p e (6, + 6> 2 ), 



From these equations we eliminate the five quantities x, y, z, p e , p a and 
the required condition that and a shall intersect, is given by the equa 
tion 

- ( 5 + 6 ), - ( 3 + 04), (! + a a ), , a (! - a 2 ) = 0. 

-(, + ,), , (! + >(), (a 3 + a 4 ), , 6(a 3 -a 4 ) 

(a 3 + 4), -(ati + Os), , (a 5 + a fi ), , c (a 5 - a 8 ) 

, (0 5 + 6 ), -(6 3 +6> 4 ), , (6, + e,}, a(0 1 -0 2 ) 

-(0 5 +0 6 ), , (0 1 + 2 \ , (6> 3 + ^ 4 ), b(0 3 -0 4 ) 

(0 3 +0<\ -& + 0J, 0,0, (^ 5 + ^ 6 ), c(0 5 -0 6 ) 

Four homogeneous equations between the co-ordinates of indicate 
that the corresponding screw lies on a certain ruled surface. Let us suppose 
that the degrees of these equations are I, m,n,r respectively, then the degree 
of the ruled surface must not exceed Slmnr. 

For express the condition that shall also intersect some given screw ct, 
we then obtain a fifth homogeneous equation containing the co-ordinates of 
in the third degree. The determination of the ratios of the six co-ordi 
nates 6-1, ... 6 is thus effected by five equations of the several degrees I, m, 
n, r, 3. For each ratio we obtain a system of values equal in number to 
the product of the degrees of the equations, i.e. to Slmnr. This is accord 
ingly a major limit to the number of points in which in general a pierces 
the surface, that is to say, it is a major limit to the degree of the surface. 
Of course we might affirm that it was the degree of the surface save for the 
possibility that through one or more of the points in Avhich a met the surface 
more than a single generator might pass. 

As an example, we may take the simple case of the cylindroid, in which 
I, m, n, r being each unity the locus is of the third degree. The screws of 

162 



244 THE THEORY OF SCREWS. [228 

a three-system which satisfy an equation of the nth degree must have as 
their locus a surface of degree not exceeding 3n. The most important 
application of this is when n = 2, in which case the screws form a quadratic 
two-system. The degree of this surface cannot exceed six, on the other 
hand, if the quadratic condition which we may write 



A0? + B0 2 * + CB? + 2F0A + 200,6, + 2H0A = 0, 

should break up into two linear factors each of these linear factors will 
correspond to a cylindroid, i.e. a surface of the third degree. Hence the 
degree of the surface must in general be neither less than six nor greater 
than six, and hence we learn that the surface which is the locus of the 
screws of a quadratic two-system is of the sixth degree. 

A particular case of special importance arises when the pitches of all the 
screws on the surface are to be the same. The statement of this condition 
is of course one equation of the second degree in the co-ordinates of the 
screw. In the case of canonical co-reciprocals, this equation would be 



But the condition that Q and a shall intersect will now submit to 
modification. We sacrifice no generality by making a of zero pitch, so 
that if 6 has a given pitch p e , the condition that a and shall intersect 
is no longer of the third degree. It is the linear equation 

2t3- ae = p e cos (a0). 

If therefore the co-ordinates of satisfy three homogeneous equations of 
degrees I, m, n respectively, in addition to the equation of the second degree 
expressing that the pitch is a given quantity, then the locus is a surface of 
degree not exceeding 2lmn. 

As the simplest illustration of this result we observe that if I, m, n be 
each unity, the locus in question is the locus of the screws of given pitch in 
a three-system. This locus cannot therefore be above the second degree, 
and we know, of course (Chapter XIV.) that the locus is a quadric. 

If I and m were each unity and if n = 2 we should then have the locus 
of screws of given pitch belonging to a four-system and whose co-ordinates 
satisfied a certain equation of the second degree. This locus is a surface of 
the fourth degree. In the special case where the given pitch is zero, the 
surface so defined is known in the theory of the linear complex. It is there 
presented as the locus of lines belonging to the complex and whose co 
ordinates further satisfy both a linear equation and a quadratic equation. 

Mr A. Panton has kindly pointed out to me that in this particular case 
the surface has two double lines which are the screws of zero pitch on the 



228] FREEDOM OF THE FOURTH ORDER. 245 

cylindroid reciprocal to the enclosing four-system. It has also eight singular 
tangent planes (four through each double line) touching all along a generator. 
The cylindroid is a very special case of this surface in which one of the 
double lines is at infinity. If the system be reciprocal to the cylindroid 
z(x 2 + y f ) + kx;y = Q, then the cylindroid to which the surface reduces is 
z(as * + y 2 ) kxy = Q. The singular tangent planes are represented by the 
two tangent planes at the limits of the cylindroid. 



CHAPTER XVII. 

FREEDOM OF THE FIFTH ORDER. 

229. Screw Reciprocal to Five Screws. 

There is no more important theorem in the Theory of Screws than that 
which asserts the existence of one screw reciprocal to five given screws. 
At the commencement, therefore, of the chapter of which this theorem is 
the foundation, it may be well to give a demonstration founded on elementary 
principles. 

Let one of the five given screws be typified by 

j tch _ 



fc Pk 7* 

while the desired screw is defined by 

(pitch = p). 



x x _y y __ z - z 



P 7 

The condition of reciprocity ( 20) produces five equations of the following 
type : 



k)7k 
From these five equations the relative values of the six quantities 

can be determined by linear solution. Introducing these values into the 
identity 

a. (yy (3z } + (az yx) 4 y (fix ay ) = 0, 

gives the equation which determines p. 



229, 230] 



FREEDOM OF THE FIFTH ORDER. 



247 



To express this equation concisely we introduce two classes of subsidiary 
magnitudes. We write one magnitude of each class as a determinant. 



= P. 



/82 



73 
74 

75 



By cyclical interchange the two analogous functions Q and R are denned. 



- y 3 Cl 3> 



73 

74 



By cyclical interchange the two analogous functions M and JV^ are denned. 
The equation for p reduces to 



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 x, y , z , whence the reciprocal screw is 
completely determined. 

For the study of the screws representing a five-system we may take the 
first screw of a set of canonical coreciprocals to be the screw reciprocal to 
the system. Then the co-ordinates of a screw in the system are 

0, 2 , 6 3 , ... 6 , 

while if X, p, v be the direction cosines of 6 and x, y, z a point thereon, and 
p e the pitch we have ( 43) 

(p e + a) cos \ z cos p + y cos v = 0. 

We can obtain at once the relation between the direction and the pitch 
of the screw belonging to the system and passing through a fixed point. 
If p e = and 2 and y be given, then the equation shows that the screw is 
limited to a plane ( 110). 

230. Six Screws Reciprocal to One Screw. 

When six screws, A l} ... A 6 are reciprocal to a single screw T, a certain 



248 THE THEORY OF SCREWS. [230 

relation must subsist between the six screws. This relation may be ex 
pressed by equating the determinant of 39 to zero. The determinant 
(which may perhaps be called the sexiant) may be otherwise expressed as 
follows : 

The equations of the screw A k are 

oc XT, y i/ 1, z ZT. , . , , , 
- - = 3 ho J2 = - - (P^ch Pk ). 
* ft 7/fc 

We shall presently show that we are justified in assuming for T the 
equations 



The condition that A k and T be reciprocal is 

77*) + ** (7& - P 



Writing the six equations of this type, found by giving k the values 
1 to 6, and eliminating the six quantities 

P<*, p&, py, <*, ft 7) 
we obtain the result : 



#i ~ a i2A> a i ft. 7i 



3/J3 + 732/3 - *3, 

4/>4 + 742/4 - ft^4, ft/>4 + a 4 ^4 ~ 744 , 74/>4 + ft4 ~ ^4 , 4, ft, 74 

a 5P5 + 75 3/5 - ftj^B, ftps + 4^5 - 75^5, 75/5 + ft^5 ~ ^S^/S, 5 , ft, 7 

6 2/6, 6 , ft, 7e 



= 0- 



By transformation to cmy parallel axes the value of this determinant is 
unaltered. The evanescence of the determinant is therefore a necessary 
condition whenever the six screws are reciprocal to a single screw. Hence 
we sacrificed no generality in the assumption that T passed through the 
origin. 

Since the sexiant is linear in x l} y 1} z i} it appears that all parallel screws 
of given pitch reciprocal to one screw lie in a plane. Since the sexiant is 
linear in a 1} ft, 7^ we have another proof of Mobius theorem ( 110). 

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. 



230] 



FREEDOM OF THE FIFTH ORDER. 



249 



(6) The six screws are members of a screw-system 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. 

For a rigid body which has freedom 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 to 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. 

Sylvester has shown* that when six lines, P, Q, R, S, T, U, are so situated 
that forces acting along them equilibrate when applied to a free rigid body, 
a certain determinant vanishes, and he speaks of the six lines so related as 
being 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 x m , y m , z m , be a point on one of the lines, the direction cosines of the 
same line being a m , /3 m ,y m , the condition is 



ii-#i7i> #i/3i-2/ii =0. 



, 72, y-fli- 

/3 3 , 73, y 3 y 3 - 

&, 74, 2/474 - 

&, 75, y 5 %- 



- #373, 



~ 2/44 



* Comptes Rendus, tome 52, p. 816. See also p. 741. 

t In our language a system of lines thus related consists of the screws of equal pitch belonging 
to a five-system. In the language of Pliicker (Neue Geometric des Raumes) a system of lines 
in involution forms a linear complex. It may save the reader some trouble to observe here 
that the word involution has been employed in a more generalised sense by Battaglini, and in 
quite a different sense by Klein. 



250 THE THEORY OF SCREWS. [230- 

It must always be possible to find a single screw X which is reciprocal 
to the six screws P, Q, R, S, T, U. Suppose the rigid body were only free 
to twist about X, then the six forces would not only collectively be in equi 
librium, 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, P, Q, R, S, T, U, which are in involution, is not necessarily perfectly 
free (Mobius). 

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 discussed by Sylvester are presented in the Theory of 
Screws 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 intersecting 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 reciprocal screw, then all 
the screws of the cylindroids form a screw system 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, due to Sylvester, is proved 
by him in a different manner. 

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. 

231. Four Screws of a Five-system on every Quadric. 

On any single sheeted hyperboloid four screws of any given pitch p can 
in general be determined which belong to any given system of the fifth 
order. A pair of these screws lie on each kind of generator. 

Let X be the screw reciprocal to the system. Take any three generators 
A, B, C of one system on the hyperboloid, and regarding them as screws of 
pitch p draw the cylindroid XA and take on this A the second screw of 
pitch p. Then the two screws of pitch p which can be drawn as transversals 
across A, B, C, A are coincident with two generators of the hyperboloid 



232] FREEDOM OF THE FJFTH ORDER. 251 

while they are also reciprocal to the cylindroid because they cross two screws 
thereon with pitches equal in magnitude but opposite in sign. They are 
therefore reciprocal to X. In like manner it can be shown that two of the 
other system of generators possess the same property. 

On every cylindroid there is as we know ( 26) one screw of a given five- 
system. This important proposition may be otherwise proved as follows. 
Let 6 be the co-ordinates of a screw on the cylindroid, then these co-ordinates 
must satisfy four linear equations. There must be a fifth equation in the 
six quantities l} ... 6 6 inasmuch as 6 is to lie on the given five-system. 
Thus from these five equations one set of values of 8 lt ... 6 6 can be 
determined. 

On a quadratic two-system ( 224) there will always be two screws 
belonging to any given five-system. For the quadratic two-system is the 
surface whose screws satisfy four homogeneous equations of which three are 
linear and one is quadratic. If another linear equation be added two 
screws on the surface can, in general, be found which will satisfy that 
equation. 

232. 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 X, as before, represent the screw reciprocal to the freedom, 
and let p be the instantaneous screw which would correspond to X as an 
impulsive screw, if the body were perfectly free ; let 77 be the screw on which 
the body receives an 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 system 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 rj. The 
movement which the body actually acquires is a twisting motion about a 
screw 6 which must lie on the cylindroid (, p). We therefore determine 6 
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 6, 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 construction. 
For by 17 the relative twist velocities about 6, , and p are known; but 
since the impulsive intensity rj " is known, the twist velocity about is 
known ( 90) ; and therefore, the twist velocity about is known ; finally, 
from the twist velocity about p, the impulsive intensity X " is determined. 



252 THE THEORY OF SCREWS. [233- 

233. Analytical Method. 

A quiescent rigid body which has freedom of the fifth order receives an 
impulsive wrench on a screw 77 : it is required to determine the instantaneous 
screw 6, about which the body will commence 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 com 
pounded with a certain wrench on X must constitute the wrench which, if 
the body were free, would make it twist about 6, whence we deduce the six 
equations (h being an unknown quantity) 



Multiplying the first of these equations by X 1} the second by X 2 , &c., adding 
the six equations thus produced, and remembering that and X are reciprocal, 
we deduce 

i/"2%A.i + X" 2V = 0. 

This equation determines X " the impulsive intensity of the reaction of 
the constraints. The co-ordinates of the required screw 6 are, therefore, 
proportional to the six quantities 



Pi Pe 

234. Principal Screws of Inertia. 

We can now determine the co-ordinates of the five principal screws of 
inertia ; for if be a principal screw of inertia, then in general 



. 

whence 



with similar values for ,, ... 6 . Substituting these values in the equation 



and making ^- = #, we have for sc the equation 

fv 






p l -co p 2 -x p 3 - x pt- a; p 5 - x p 6 - x 
This equation ts of the fifth degree, corresponding to the five principal 
screws of inertia. If x denote one of the roots of the equation, then the 
corresponding principal screw of inertia has co-ordinates proportional to 
Xj X-s X3 X^ _ X 5 X 6 

^~- x" PI-X" p 3 -x" pi-" Ps-x" PS- 



235] FREEDOM OF THE FIFTH ORDER. 253 

We can easily verify as in 84 that these five screws are co-reciprocal and 
are also conjugate screws of inertia. 

It is assumed in the deduction of this quintic that all the quantities 
XJ...XB are different from zero. If one of the quantities, suppose X l5 had 
been zero this means that the first absolute principal Screw of Inertia 
would belong to the w-system expressing the freedom. 

Let us suppose that Xj = then the equations are 



Of course one solution of this system will be V" = 0, , = ... = 0. This 
means that the first absolute principal Screw of Inertia is also one of the 
principal Screws of Inertia in the rc-system, as should obviously be the case. 
For the others = and we have an equation of the fourth degree in 



, 



P* - % 

In the general case we can show that there are no imaginary roots in the 
quintic, for since the screws 

^1 ^-2 X 



and 



p 1 - x p 2 - x p s - x" 
are conjugate screws of inertia, we must have (81) 



If x = a. + if$ ; y = a ift then this equation reduces to 

v frV _ 



but as these are each positive terms their sum cannot be zero. This is a 
particular case of 86. (See Appendix, note 2.) 

235. The limits of the roots. 

We can now show the limits between which the five roots, just proved to 
be real, must actually lie in the equation 



254 THE THEORY OF SCREWS. [235, 

11 11 

substitute p , = . = . 6 = x = -\ 

ft ?> ft y 

and suppose qi, q z , qa, q*, q*, q& to be in descending order of magnitude. 

"X 2 -\ 2 "\ 2 

Thus -JSL + _*_ + . ..+_*_=<). 

y - ?i y - & y - ft 



That is v(y - ft)(y - ?,)(y - ? 4 ) (y -?5)(y - &) + 

+ V (y - ?0 (y - 9-2) (y - &) (y - ? 4 ) (y - ? B ) = 0. 

In the left-hand member of this equation substitute the values 

qi, q*, q a , ?4> q*, q* 

successively for y ; five of the six terms vanish in each case, and the values of 
the remaining term (and therefore of the whole member) are alternately 
positive and negative. 

The five values of y must therefore lie in the intervals between the six 
quantities q lt q 2 , ... q s , the roots are accordingly proved to be real and distinct 
(unless one of the quantities \ 1} \ 2 , X 3 , \ 4 , \ 5 , X 6 = and a further condition 
hold, or unless some of the quantities q 1} ... q e be equal). 

The values of p 1} ... p 6 are a, b, c; and we suppose a, b, c, positive 
and a > b > c. 

The values of y lie in the successive intervals between 

1 1 1 _1 _1 _1 

c b a a b c 

and consequently of the roots of the equation in x. 

Two are positive and lie between a and b, and between b and c respectively. 

Two are negative and lie between a and b, and between b and c 
respectively. 

The last is either positive and > a or negative and < a. 

236. The Pectenoid. 

A surface of some interest in connection with the freedom of the fifth 
order may be investigated as follows. 

Let a be the pitch of the one screw &>, to which the five system is 
reciprocal. 

Take any point on o> and draw through any two right lines OF, and 
OZ which are at right angles and which lie in the plane perpendicular 
to a). 



236] 



FREEDOM OF THE FIFTH ORDER. 



255 



Then if 6 be a screw of the five-system with direction cosines cos X, 
cos //,, cos v, and if as, y, z be a point on the screw 6 and p e its pitch we must 
have ( 216) 

( p e + a) cos X + z cos /* y cos v = 0. 




Fig. 41. 



Everything that we wish to specify about the five-system may be 
conveniently inferred from this equation. 

For example, let it be desired to find the locus of the screws of a five- 
system which can be drawn through a given point as , y , z and have the 
given pitch p e . 



We have 



( p e + a) cos \-\-z cos p y cos v = 0. 



If os, y, z be a point on 6 we may substitute x x, y y, z z for cos X, 
cos ft, cos v, and we obtain 



whence we see that the locus is a plane, as has been already proved other 
wise. (When the pitch is zero, this is Mb bius theorem, 110.) 

If we change the origin to some other point P which may with complete 
generality be that point whose co-ordinates are o, h, o and call X, Y, Z the 
co-ordinates with these new axes, the equation becomes 

(p e + a)c 



256 THE THEORY OF SCREWS. [236 

Let the radius vector of length p g + a = R be marked off along each screw 
6 drawn through P, then the equation becomes 



or squaring (Z 2 + F 2 + Z*) X* = h*Z\ 

This represents a surface of the fourth degree. A model of the surface 
has been constructed*. It is represented in Fig. 41, and from its resemblance 
to the valves of a scallop shell the name pectenoid is suggested. 

The geometrical nature of a pectenoid is thus expressed. Given a screw 
a of pitch p a and a point situated anywhere. If a screw 6 drawn through 
be reciprocal to a then the extremity of a radius vector from along 6 equal 
to p a + p e will trace out a pectenoid. 

All pectenoids are similar surfaces, they merely differ in size in accordance 
with the variations of the quantity h. The perpendicular from upon a is 
a nodal line, and this is the only straight line on the surface. The pectenoid 
though unclosed is entirely contained between the pair of parallel planes 
Z = + h, Z = h. Sections parallel to the plane of Z are hyperbolas. Any 
plane through the nodal line cuts the pectenoid in a circle. 

A straight wire at right angles to the nodal line marked on the model 
indicates the screw reciprocal to the five-system. A second wire starts 
from the origin and projects from the surface. It is introduced to show 
concisely what the pectenoid expresses. If this wire be the axis of a 
screw 6 whose pitch when added to the pitch of the screw a, is equal to 
the intercept from the origin to the surface then the two screws are 
reciprocal. The interpretation of the nodal line is found in the obvious 
truth that when two screws intersect at right angles they are reciprocal 
whatever be the sum of their pitches. One of the circular sections made 
by a plane drawn through the nodal line is also indicated in the model. The 
physical interpretation is found in the theorem already mentioned, that all 
screws of the same pitch drawn through the same point and reciprocal to 
a given screw will lie in a plane. 

With the help of the pectenoid we can give another proof of the theorem 
that all the screws of a four-system which can be drawn through a point lie 
on a cone of the second degree ( 123). 

Let be the point and let a and /3 be two screws on the cylindroid 
reciprocal to the system. Let 6 be a screw through belonging to the 
four-system and therefore reciprocal to a and to /9. 

Then for the pectenoid relating to and a, we have 

(p e + pa )M-kN=0, 
where M=0, N=0 represent planes passing through 0. 

* Transaction* of the Royal Irish Academy, Vol. xxv. Plate xn. (1871). 



236] FREEDOM OF THE FIFTH ORDER. 

In like manner for the pectenoid relating to and /3, we have 



257 



where M = 0, N = also represent planes passing through 0, whence 
eliminating p e we have 

(p a -pe) MM - kNM + k N M= 0. 
The equation of the pectenoid can also be deduced directly as follows. 

Let five screws of the five-system be given. Take a point on one of 
these screws (a) and through draw four screws /3, 7, 8, e which belong to 
the four-system defined by the remaining four screws of the original five. 
Let there be any three rectangular axes drawn through 0. Let ct l , 2 , a 3 be 
the direction cosines of a and let /3 1( /3 2 , /3 3 be the direction cosines of /?, 
and similarly for 7, 8, e. Let 6 be some other screw of the five-system 
which passes through and let 1} 6 2 , 3 be its direction cosines, then if 
twists of amplitudes a, ft , 7 , 5 , e , & neutralize we must have 



77i 



ee 2 



a 3 + + 773 + 3 + ee s 
because the rotations neutralize, and also 



a.p a a. 2 



whence by elimination of a , /3 , . . . 6 , we have 
a 2 @ 2 72 S 2 e 2 

83 >3 73 63 63 



i3 



2 = 0, 
3 = 0, 



f p 9 1 = 0, 
6 p e 6 z = 0, 
p e & 3 = 0, 



= 0. 



This equation has the form 



Pe L e. + M e. + N e, 

Let p e = p /i and l = x-^- p, 6 z = y-^p, 3 = z H- p then by reduction and 
transformation of axes we obtain 



where y and 2 are planes at right angles and a is constant. This is the 
equation of the pectenoid. 

17 



a 



CHAPTER XVIII. 

FREEDOM OF THE SIXTH ORDER. 

237. Introduction. 

When a rigid body has freedom of the sixth order, it is perfectly free. 
The screw system of the sixth order includes every screw in space. The 
statement that there is no reciprocal screw to such a system is merely a 
different way of asserting the obvious proposition that when a body is 
perfectly free it cannot remain in equilibrium, if the forces which act upon 
it have a resultant. 

238. Impulsive Screws. 

Let A lt A 2 , ... denote a series of instantaneous screws which correspond 
respectively to the impulsive screws R lt R 2 , ... the body being perfectly 
free. Corresponding to each pair A 1} RI is a certain specific parameter. 
This parameter may be conveniently defined to be the twist velocity pro 
duced about A! by an impulsive wrench on R^, of which the intensity is one 
unit. If six pairs, A lt jR T ; A a , R. 2 , ... be known, and also the corresponding 
specific parameters, then the impulsive wrench on any other screw R can 
be resolved into six impulsive wrenches on R l ,...R 6 , these will produce 
six known twist velocities on A 1} ... A G , which being compounded determine 
the screw A, the twist velocity about A, and therefore the specific para 
meter 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 determine 
completely the effect of any other impulsive wrench. 

If seven pairs of corresponding instantaneous and impulsive screws be 
given, then the relation between every other pair is absolutely determined. 
It appears from 28 that appropriate twist velocities about A^,...A 7 can 
neutralise. When this is the case, the corresponding impulsive wrenches 
on R^^.R?, must equilibrate, and therefore the relative values of the 
intensities are known. It follows that the specific parameter of each pair 
At, RI is proportional to the quotient obtained by dividing the sexiant of 



237-239] FREEDOM OF THE SIXTH ORDER, 259 

A 2 ,...A R , by the sexiant of R. 2 ,...R 6 . With the exception of a common 
factor, the specific parameter of every pair of screws is therefore known, 
when seven corresponding screws are known. It will be shown in Chap. XXI. 
that three corresponding pairs are really sufficient. 

When seven instantaneous screws are known, and the corresponding 
seven impulsive screws, we are therefore enabled by geometrical construction 
alone to deduce the instantaneous screw corresponding to any eighth impulsive 
screw and vice versa. 

A precisely 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 system of forces which have a potential, and if the twists about seven 
given screws evoke wrenches about seven other given screws, then, without 
further information 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 may present the results of the present section in another form. We 
must conceive two corresponding systems of screws, of which the correspond 
ence is completely 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 correspond another screw viewed as belonging to the other system. 
Six screws can be found, each of which coincides with its correspondent. 
To a screw system of the nth order and rath degree in one system will 
correspond a screw system of the nth order and rath degree in the other 
Sj stem. 

We add here a few examples to illustrate the use which may be made of 
screw co-ordinates. 

239. 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 f}i,...i] 6 be the co-ordinates of the force referred to the absolute 
principal screws of inertia, then ( 35) 



and from ( 41) it follows that the direction cosines of 77 with respect to the 
principal axes through the centre of inertia are 



172 



260 THE THEORY OF SCREWS. [239- 

If a, b, c be the radii of gyration, then the instantaneous screw cor 
responding to i] has for co-ordinates 

, % _fh ,Va _ r h ,^5 _^s 

I y y I T > 71 I i 

a a o o c c 

The condition that 77 and its instantaneous screw shall be parallel to a 
pair of conjugate diameters of the momental ellipsoid is 



or 

But if the impulsive wrench on 77 be a force, then the pitch of 77 is zero, 
whence the theorem is proved. 

240. 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 impulsive screw. 

Let i)!, ... i) 6 be the axis of the rotation, then 

2^ V = 0, 
or 

a (77! - 7; 2 ) (77! + O + b (773 - 774) (773 + 774) + c (775 - 77 6 ) (775 + 77 6 ) = 0, 

whence the screw of which the co-ordinates are + ar) l , arj. 2 , + brj 3 , ... is 
perpendicular to 77, 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. 

241. Principal Axis. 

If 77 be a principal axis of a rigid body, it is required to prove that 

SjH*y0, 
reference being made to the absolute principal screws of inertia. 

For in this case a force along a line Q intersecting 77, compounded with 
a couple in a plane perpendicular to 77, must constitute an impulsive wrench 
to which 77 corresponds as an instantaneous screw, whence we deduce ( 120), 
h and k being the same for each coordinate, 

., h dR 



h dR 



242] FKEEDOM OF THE SIXTH OIIDER. 261 

Expressing the condition that p e = 0, we have 



but we have already seen ( 131) 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 



This shows that if the body were free, then an impulsive force suitably 
placed would make the body commence to rotate about 77. 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. 

It may also be asked at what point of the body one of the three 
principal axes coincides with 77 ? This point is the intersection of 6 and 77. 
To determine the co-ordinates of 6 it is only necessary to find the relation 
between h and k, and this is obtained by expressing the condition that 6 is 
reciprocal to 77, whence we deduce 

Zh + kuj = 0. 

Thus is known, and the required point is determined. If the body be 
fixed at this point, and then receive the impulsive couple perpendicular to 
77, the instantaneous reaction of the point will be directed along 0. 

242. Harmonic Screws. 

We shall conclude by stating for the sixth order the results which are 
included as particular cases of the general theorems in Chapter IX. 

If a perfectly free rigid body be in equilibrium under the influence of a 
conservative system of forces, then six screws can generally 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 circumstances, the movement of the body is com 
pounded of twist oscillations about the six harmonic screws. 

* Townsend, Educational Times Reprint, Vol. xxi. p. 107. 



CHAPTER XIX. 

HOMOGRAPHIC SCREW SYSTEMS*. 

243. Introduction. 

Several of the most important parts of the Theory of Screws can be 
embraced in a more general theory. I propose in the present chapter to 
sketch this general theory. It will be found to have points of connexion 
with the modern higher geometry ; in particular the theory of Homographic 
Screws is specially connected with the general theory of correspondence. I 
believe it will be of some interest to show how these abstract geometrical 
theories may be illustrated by dynamics. 

244. On Plane Homographic Systems. 

It may be convenient first to recite the leading principle of the purely 
geometrical theory of homography. We have already had to mention a 
special case in the Introduction. 

Let a be any point in a plane, and let ft be a corresponding point. Let 
us further suppose that the correspondence is of the one-to-one type, so that 
when one a is given then one ft is known, when one ft is given then it is the 
correspondent of a single a. The relation is not generally interchangeable. 
Only in very special circumstances will it be true that ft, regarded as in the 
first system, will correspond to a in the second system. 

The general relation between the points a and ft can be expressed by the 
following equations, where a 1} a. 2 , 3 are the ordinary trilinear co-ordinates of 
a, and J3 1} /3 a , fi 3 , the co-ordinates of ft, 

ft = (11)0! +(12)0, + (13) a,, 



ft = (31) 0^(32) a, + (33) 03. 

In these expressions (11), (12), &c., are the constants defining the particular 
character of the homographic system. 

* Proc. Hoy. Irish Acad. Ser. n. Vol. HI. p. 435 (1881). 



243-246] iioMOGiiAPHic SCREW SYSTEMS. 263 

There are in general three points, which coincide with their corre 
spondents. These are found by putting 

& = Pi ; ft* = pa., ; j3 3 = pa a . 

Introducing these values, and eliminating a l , a. 2> a s , we obtain the following 
equation for p : 

0-| (11) -p, (12), (13) 

(21), (22) -p, (23) 

(31), (32), (33) -p 

If we choose these three points of the vertices of the triangle of reference, 
the equations relating y with x assume the simple form, 

& = /i i ; & =/ 2 <* 2 ; & =/ 3 a , 

where /!,/ 2 ,/ 3 are three new constants. 

245. Homographic Screw Systems. 

Given one screw a, it is easy to conceive that another screw ft correspond 
ing thereto shall be also determined. We may, for example, suppose that 
the co-ordinates of ft ( 34) shall be given functions of those of a. We might 
imagine a geometrical construction by the aid of fixed lines or curves by 
which, when an a is given, the corresponding ft shall be forthwith known : 
again, we may imagine a connexion involving dynamical conceptions such as 
that, when a is the seat of an impulsive wrench, ft is the instantaneous screw 
about which the body begins to twist. 

As a moves about, so the corresponding screw ft will change its position 
and thus two corresponding screw systems are generated. Regarding the 
connexion between the two systems from the analytical point of view, the 
co-ordinates of a and ft will be connected by certain equations. If it be 
invariably true that a single screw ft corresponds to a single screw a, and that 
conversely a single screw a corresponds to a single screw ft ; then the two 
systems of screws are said to be homographic. 

A screw a. in the first system has one corresponding screw ft in the 
second system ; so also to ft in the first system corresponds one screw 
in the second system. It will generally be impossible for a and a to coincide, 
but cases may arise in which they do coincide, and these will be discussed 
further on. 

246. Relations among the Co-ordinates. 

From the fundamental property of two homographic screw systems the 
co-ordinates of ft must be expressed by six equations of the type 



264 THE THEORY OF SCREWS. [246- 

If these six equations be solved for Oj, ... otg we must have 



a, =., .... 

As a single a is to correspond to a single /3, and vice versa, these equations 
must be linear : whence we have the following important result : 

In two homographic screw systems the co-ordinates of a screw in one system 
are linear functions with constant coefficients of the co-ordinates of the corre 
sponding screw in the other system. 

If we denote the constant coefficients by the notation (11), (22), &c., then 
we have the following system of equations : 

13, = (11) a, + (12) a, + (13) o 3 + (14) 4 + (15) a 5 + (16) 6 , 
& = (21) a + (22) as + (23) a 3 + (24) 4 + (25) a 5 + (26) a,, 

& = (61) a, + (62) a, + (63) a 3 + (64) a 4 + (65) a 5 + (66) cv 

247. The Double Screws. 

It is now easy to show that there are in general six screws which coincide 
with their corresponding screws; for if ^ 1 = p%i, j3 2 = pa. 2 , &c., we obtain an 
equation of the sixth degree for the determination of p. We therefore 
have the following result : 

In two homographic screw systems six screws can in general be found, each 
of which regarded as a screw in either system coincides with its correspondent 
in the other system. 

248. The Seven Pairs. 

In two homographic rows of points we have the anharmonic ratio of 
any four points equal to that of their correspondents. In the case of two 
homographic screw systems we have a set of eight screws in one of the 
systems specially related to the corresponding eight screws in the other 
system. 

We first remark that, given seven pairs of corresponding screws in the two 
systems, then the screw corresponding to any other given screw is deter 
mined. For from the six equations just written by substitution of known 
values of ail, ... 6 and /3 1; ... /3 6 , we can deduce six equations between (11), 
(12), &c. As, however, the co-ordinates are homogeneous and their ratios are 
alone involved, we can use only the ratios of the equations so that each pair 
of screws gives five relations between the 36 quantities (11), (12), &c. The 



249] HOMOGRAPHIC SCREW SYSTEMS. 265 

seven pairs thus give 35 relations which suffice to determine linearly the 
ratios of the coefficients. The screw y9 corresponding to any other screw a is 
completely determined ; we have therefore proved that 

When seven corresponding pairs of screws are given, the two homographic 
screw systems are completely determined. 

A perfectly general way of conceiving two homographic screw systems 
may be thus stated : Decompose a wrench of given intensity on a screw a 
into wrenches on six arbitrary screws. Multiply the intensity of each of the 
six component wrenches by an arbitrary constant ; construct the wrench on 
the screw /3 which is the resultant of the six components thus modified; 
then as a moves into every position in space, and has every fluctuation in 
pitch, so will /3 trace out the homographic screw system. 

It is easily seen that in this statement we might have spoken of twist 
velocities instead of wrenches. 

249. Homographic n-systems. 

The seven pairs of screws of which the two systems are defined cannot be 
always chosen arbitrarily. If, for example, three of the screws were co- 
cylindroidal, then the three corresponding screws must be co-cylindroidal, 
and can only be chosen arbitrarily subject to this imperative restriction. 
More generally we shall now prove that if any n + 1 screws belong to an 
n -system (69), then the n + 1 corresponding screws will also belong to an 
n-system. If n + 1 screws belong to an ?i-system it will always be possible to 
determine the intensities of certain wrenches on the n + 1 screws which when 
compounded together will equilibrate. The conditions that this shall be 
possible are easily expressed. Take, for example, n = 3, and suppose that 
the four screws a, fi, 7, 8 are such that suitable wrenches on them, or twist 
velocities about them, neutralize. It is then obvious ( 76) that each of the 
determinants must vanish which is formed by taking four columns from 
the expression 

i, -,, o 3> a,, a 5 , 2 6 



76 



It is, however, easy to see that these determinants will equally vanish for 
the corresponding screws in the homographic system ; for if we take as screws of 
reference the six common screws of the two systems, then we have at once 
for the co-ordinates of the screw corresponding to a 

(ll)a lf (22)0,, (33) a,, (44) 4 , (55) a., (66)*. 



2GG THE THEORY OF SCREWS. [249- 

Whcri these substitutions are made in the determinants it is plain that 
they still vanish ; we hence have the important result that 

The screws corresponding homographically to the screws of an n-system 
form another n-system. 

Thus to the screws on a cylindroid will correspond the screws on a 
cylindroid. It is, however, important to notice that two reciprocal screws 
have not in general two reciprocal screws for their correspondents. We thus 
see that while two reciprocal screw systems of the nth and (6 ?i)th orders 
respectively have as correspondents systems of the same orders, yet that 
their connexion as reciprocals is divorced by the homographic transforma 
tion. 

Reciprocity is not, therefore, an invariantive attribute of screws or screw 
systems. There are, however, certain functions of eight screws analogous to 
anharmoriic ratios which are invariants. These functions are of considerable 
interest, and they are not without physical significance. 

250. Analogy to Anharmonic Ratio. 

We have already ( 230) discussed the important function of six screws 
which is called the Sexiant. This function is most concisely written as the 
determinant (a^^Js^s^s) where a, /3, 7, B, e, are the screws. In Sylvester s 
language we may speak of the six screws as being in involution when their 
sexiant vanishes. Under these circumstances six wrenches on the six screws 
can equilibrate ; the six screws all belong to a 5-system, and they possess one 
common reciprocal. In the case of eight screws we may use a very concise 
notation; thus 12 will denote the sexiant of the six screws obtained by 
leaving out screws 1 and 2. It will now be easy to show that functions of the 
following form are invariants, i.e. the same in both systems: 

12 . 34 

13. 24 

It is in the first place obvious that as the co-ordinates of each screw enter to 
the same degree in the numerator and the denominator, no embarrassment 
can arise from the arbitrary common factor with which the six co-ordinates of 
each screw may be affected. In the second place it is plain that if we replace 
each of the co-ordinates by those of the corresponding screw, the function 
will still remain unaltered, as all the factors (11), (22), &c., will divide out. We 
thus see that the function just written will be absolutely unaltered when 
each screw is changed into its corresponding screw. 

By the aid of these invariant functions it is easy, when seven pairs of 
screws are given, to construct the screw corresponding to any given eighth 



252] HOMOGRAPHIC SCREW SYSTEMS. 2G7 

screw. We may solve this problem in various ways. One of the simplest 
will be to write the five invariants 

12.38 13.48 14\58 15.68 16.78 
13.28 14.38 15.48 16.58 IT . 68* 

These can be computed from the given eight screws of one system ; hence 
we have five linear equations to determine the ratios of the coefficients of the 
required eighth screw of the other system. 

It would seem that of all the invariants of eight screws, five alone can 
be independent. These five invariants are attributes of the eight-screw 
system, in the same way that the anharmonic ratio is an attribute of four 
collinear points. 

251. A Physical Correspondence. 

The invariants are also easily illustrated by considerations of a me 
chanical nature. To a wrench on one screw corresponds a twist on the 
corresponding screw, and the ratio of the intensities of the wrench and twist 
is to be independent of those intensities. We may take a particular case to 
illustrate the argument : Suppose a free rigid body to be at rest. If that 
body be acted upon by an impulsive system of forces, those forces will 
constitute a wrench on a certain screw a. In consequence of these forces the 
body will commence to move, and its instantaneous motion cannot be 
different from a twist velocity about some other screw /3. To one screw a 
will correspond one screw 0, and (since the body is perfectly free) to one 
screw /3 will correspond one screw a. It follows, from the definition of homo- 
graphy, that as a. moves over every screw in space, ft will trace out an homo- 
graphic system.... From the laws of motion it will follow, that if F be the 
intensity of the impulsive wrench, and if V be the twist velocity which that 
wrench evokes, then F~ V will be independent of F and V, though, of course, 
it is not independent of the actual position of a and /3. 

252. Impulsive and Instantaneous Systems. 

It is known ( 230) that when seven wrenches equilibrate (or when 
seven twist velocities neutralize), the intensity of the wrench (or the twist 
velocity) on any one screw must be proportional to the sexiant of the six non- 
corresponding screws. 

Let F 1S , F., s , ... F 7S be the intensities of seven impulsive wrenches on the 
screws 1, 2, ... 7, which equilibrate, then we must have 



18 28 78 



268 THE THEORY OF SCREWS. [252- 

Similarly, by omitting the first screw, we can have seven impulsive wrenches 
which equilibrate, where 



18 



hence we have 



13.28 FK.F& 

Let the instantaneous twist velocity corresponding to F 1S be denoted by 
F 18 , then, as when seven wrenches equilibrate, the seven corresponding twist 
velocities must also equilibrate, we must have in the corresponding system, 

12. 38 = VnV* 

13.28 ViV 

But we must have the twist velocity proportional to the impulsive intensity ; 
hence, from the second pair of screws we have 



and from the third pair, 

F V F V 

^38 38 * 13 13 ) 

hence we deduce 

VlZ ^38 _ ^12 ff 



1^13 ^28 -^13 ^28 

and, consequently, the function of the eight impulsive screws 

12.38 

13.28 
must be identical with the same function of the instantaneous screws. 

It should, however, be remarked, that the impulsive and instantaneous 
screws do not exhibit the most general type of two homographic systems. A 
more special type of homography, and one of very great interest, characterizes 
the two sets of screws referred to. 

253. Special type of Homography. 

If the general linear transformation, which changes each screw a into its 
correspondent 6, be specialized by the restriction that the co-ordinates of 6 
are given by the equations 

fi- 1 

"i 

Pi 



I <* 

6 da 6 



254] HOMOGRAPHIC SCEEW SYSTEMS. 269 

where U is any homogeneous function of the second order in a lt ...a 6 , and 
where p lt ...p 6 are the pitches of the screws of reference, then the two 
systems are related by the special type of homography to which I have 
referred. 

The fundamental property of the two special homographic systems is 
thus stated : 

Let a and /3 be any two screws, and let 6 and <j> be their correspondents, 
then, when a is reciprocal to <j>, /3 will be reciprocal to 9. 

We may, without loss of generality, assume that the screws of reference 
are co-reciprocal, and in this case the condition that ft and 6 shall be co- 
reciprocal is 

= ; 



but by substituting for 1} ... 6 , this condition reduces to 

dU 



Similarly, the condition that a and < shall be reciprocal is 

dU dU 

~ = 



It is obvious that as U is a homogeneous function of the second degree, 
these two conditions are identical, and the required property has been 
proved. 

254. Reduction to a Canonical form. 

It is easily shown that by suitable choice of the screws of reference the 
function U may, in general, be reduced to the sum of six square terms. We 
now proceed to show that this reduction is generally possible, while still 
retaining six co-reciprocals for the screws of reference. 

The pitch p a of the screw a is given by the equation ( 38), 



the six screws of reference being co-reciprocals, the function p a must retain 
the same form after the transformation of the axes. The discriminant of 
the function 



equated to zero will give six values of X ; these values of X will determine 

the coefficients of U in the required form. I do not, however, enter further 

into the discussion of this question, which belongs to the general theory of 
linear transformations. 



270 THE THEORY OF SCREWS. [254- 

The transformation having been effected, an important result is im 
mediately deduced. Let the transformed function be denoted by 



then we have 

ft=- 
PI 



PI 

whence it appears that the six screws of reference are the common screws of 
the two systems. We thus find that in this special case of homography 

The six common screws of the two systems are co-reciprocal. 

The correspondence between impulsive screws and instantaneous screws 
is a particular case of the type here referred to. The six common screws of 
the two systems are therefore what we have called the principal screws of 
inertia, and they are co-reciprocal. 

255. Correspondence of a Screw and a system. 

We shall sometimes have cases in which each screw of a system cor 
responds not to a single screw but to a system of screws. For the sake of 
illustration, suppose the case of a quiescent rigid body with two degrees of 
freedom and let this receive an impulsive wrench on some screw situated 
anywhere in space. The movement which the body can accept is limited. 
It can, indeed, only twist about one of the singly infinite number of screws, 
which constitute a cylindroid. To any screw in space will correspond one 
screw on the cylindroid. But will it be correct to say, that to one screw on 
the cylindroid corresponds one screw in space ? The fact is, that there are 
a quadruply infinite number of screws, an impulsive wrench on any one of 
which will make the body choose the same screw on the cylindroid for its 
instantaneous movement. The relation of this quadruply infinite group is 
clearly exhibited in the present theory. It is shown in 128 that, given a 
screw a on the cylindroid, there is, in general one, but only one screw 6 on 
the cylindroid, an impulsive wrench on which will make the body commence 
to twist about a. It is further shown that any screw whatever which fulfils 
the single condition of being reciprocal to a single specified screw on the 
cylindroid possesses the same property. The screws corresponding to a thus 
form a five-system. The correspondence at present before us may therefore 
be enunciated in the following general manner. 

To one screw in space corresponds one screw on the cylindroid, and to one 
screw on the cylindroid corresponds a five-system in space. 



257] HOMOGRAPHIC SCREW SYSTEMS. 271 

256. Correspondence of m and n systems . 

We may look at the matter in a more general manner. Consider an 
w-system (J.) of screws, and an w-system (B) (m>n). (If we make ra = 6 
and n = 2, this system includes the system we have been just discussing.) 
To one screw in A will correspond one screw in B, but to one screw in B 
will correspond, not a single screw in A, but an (m + 1 ?i)-system of screws. 

If m = n, we find that one screw of one system corresponds to one screw 
of the other system. Thus, if m = n = 2, we have a pair of cylindroids, and 
one screw on one cylindroid corresponds to one screw on the other. If 
m = 3, and n = 2, we see that to each screw on the cylindroid will cor 
respond a whole cylindroid of screws belonging to the three-system. For 
example, if a body have freedom of the second order and a screw be indicated 
on the cylindroid which defines the freedom, then a whole cylindroid full of 
screws can always be chosen from any three-system, an impulsive wrench on 
any one of which will make the body commence to twist about the indicated 
screw. 

257. Screws common to the two systems. 

The property of the screws common to the two homographic systems 
will of course require some modification when we are only considering an 
wi-system and an ?i-system. Let us take the case of a three-system on the 
one hand, and a six-system, or all the screws in space, on the other hand. 
To each screw a of the three-system A must correspond, a four-system, B, 
so that a cone of the screws of this four-system can be drawn through every 
point in space. It is interesting to note that one screw /3 can be found, 
which, besides belonging to B, belongs also to A. Take any two screws 
reciprocal to B, arid any three screws reciprocal to A, then the single screw 
/3, which is reciprocal to the five screws thus found, belongs to both A and 
B. We thus see that to each screw a of A, one corresponding screw in the 
same system can be determined. The result just arrived at can be similarly 
shown generally, and thus we find that when every screw in space cor 
responds to a screw of an ?i-system, then each screw of the n-system will 
correspond to a (7 ?i)-system, and among the screws of this system one 
can always be found which lies on the original n-system. 

As a mechanical illustration of this result we may refer to the theorem 
( 96), that if a rigid body has freedom of the nth order, then, no matter 
what be the system of forces which act upon it, we may in general combine 
the resultant wrench with certain reactions of the constraints, so as to 
produce a wrench on a screw of the n-system which defines the freedom of 
the body, and this wrench will be dynamically equivalent to the given 
system of forces. 



272 THE THEORY OF SCREWS. [258, 

258. Corresponding Screws defined by Equations. 

It is easy to state the matter analytically, and for convenience we shall 
take a three-system, though it will be obvious that the process is quite 
general. 

Of the six screws of reference, let three screws be chosen on the three- 
system, then the co-ordinates of any screw on that system will be a 1( 2 , 3 , 
the other three co-ordinates being equal to zero. The co-ordinates of the 
corresponding screw ft must be indeterminate, for any screw of a four-system 
will correspond to ft. This provision is secured by /3 4) /3 5 , /3 f) remaining quite 
arbitrary, while we have for @ lt j3 2 , @ 3 the definite values, 



If we take /3 4 , /3 5 , /3 6 all zero, then the values of ($ l} /3 2 , /3 3 , just written, give 
the co-ordinates of the special screw belonging to the three-system, which 
is among those which correspond to a. 

As a moves over the three-system, so will the other screw of that system 
which corresponds thereto. There will, however, be three cases in which the 
two screws coincide ; these are found at once by making 

Pi = p<*i , /3 2 = p* 2 ; /3 3 = pa 3 , 
whence we obtain a cubic for p. 

It is thus seen that generally n screws can be found on an ?z-system, so 
that each screw shall coincide with its correspondent. As a dynamical 
illustration we may give the important theorem, that when a rigid body 
has n degrees of freedom, then n screws can always be found, about any 
one of which the body will commence to twist when it receives an impulsive 
wrench on the same screw. These screws are of course the principal screws 
of inertia ( 84). 

259. Generalization of Anharmonic Ratio. 

We have already seen the anharmonic equality between four screws on a 
cylindroid, and the four corresponding screws ; we have also shown a quasi 
anharmonic equality between any eight screws in space and their cor 
respondents. More generally, any n + 2 screws of an w-system are connected 
with their n + 2 correspondents, by relations which are analogous to an 
harmonic properties. The invariants are not generally so simple as in the 
eight-screw case, but we may state them, at all events, for the case of n = 3. 

Five screws belonging to a three-system, and their five correspondents 



259] HOMOGKAPHIC SCREW SYSTEMS. 273 

are so related, that when nine are given, the tenth is immediately deter 
mined ; for this two data are required, that being the number required to 
specify a screw already known to belong to a given three-system. 

We may, as before, denote by 12 the condition that the screws 3, 4, 5 
shall be co-cylindroidal. This, indeed, requires no less than four distinct 
conditions, yet, as pointed out ( 76), functions can be found whose evanes 
cence will supply all that is necessary. Nor need this cause any surprise, 
when it is remembered that the evanescence of the sine of an angle between 
two lines contains the two conditions necessary that the direction cosines are 
identical. The function 

12.34 
13.24 

can then be shown to be an invariant which retains its value unaltered when 
we pass from one set of five screws in a three-system to the corresponding 
set in the other system. 



B - 18 



CHAPTER XX. 



EMANANTS AND PITCH INVARIANTS. 

260. The Dyname. 

If we wish to speak of a magnitude which may be a twist or a wrench or 
a twist velocity it is convenient to employ the word Dyname used by 
Pliicker* and by other writers. The Dyname a is completely expressed by 
its components a l} ... a 6 n the six screws of reference. These six quantities 
are quite independent. They may be considered as the co-ordinates of the 
Dyname. 

Let a be the intensity of the Dyname on a ; then of is a factor in each of 
!, ... a 6 , and if the Dyname be replaced by another on the same screw 
a, but of intensity #a , the co-ordinates of this new Dyname will be 

#!, ... XO.Q. 

Let ft be a second Dyname on another screw quite arbitrary as to its 
position and as to its intensity ft . Let the co-ordinates of ft, referred to the 
same screws of reference, be ft 1} ... /3 6 . If we suppose a Dyname of intensity 
yft on the screw ft, then its co-ordinates will be yft 1} ... yft 6 . Let us now 
compound together the two Dynames of intensities oca and yft on the screws 
a. and ft. They will, according to the laws for the composition of twists and 
wrenches ( 14), form a single Dyname on a third screw lying on the same 
cylindroid as a and ft. The position of the resultant screw is such that it 
divides the angle between a and ft into parts whose sines have the ratio of y 
to x. The intensity of the resultant Dyname is also determined (as in the 
parallelogram of force) to be the diagonal where x and y are the sides, and 
the angle between them is the angle between a and ft. It is important to 
notice that in the determination of this resultant the screws to which the co 
ordinates are referred bear no part ; the position of the resultant Dyname on 
the cylindroid as well as its intensity each depend solely upon the two 
original Dynames, and on the numerical magnitudes x and y. 

* Pliicker, Fundamental views regarding Mechanics, Phil. Trans. 1866, Vol. CLVI. pp. 361 
380. 



260, 261] EMANANTS AND PITCH INVARIANTS. 275 

We have now to form the co-ordinates of the resulting Dyname, or its 
components when decomposed along the six screws of reference. The first 
Dyname has a component of intensity x^ on the first screw ; and as the 
second Dyname has a component y^ 1} it follows that the sum of these two 
must be the component of the resultant. Thus we have for the co-ordinates 
of the resultant Dyname the expressions 



261. Emanants. 

Let us suppose that without in any particular altering either of the 
Dynames a and ft we make a complete change of the six screws of reference. 
Let the co-ordinates of a with regard to these new screws be \, ... \ 6 , and 
those of (3 be //, 1} ... //,. Precisely the same argument as has just been used 
will show that the composition of the Dynames xo! and yfi will produce a 
Dyname whose co-ordinates are x\^ + y/j, lt . . . x\ 6 + y(j, s . We thus see that the 
Dyname defined by the co-ordinates x^ + yfti, ... xa 6 + y{3 6 , referred to the 
first group of reference screws is absolutely the same Dyname as that defined 
by the co-ordinates X,j + y^, ... x\ 6 + yv 6 referred to the second group 
of reference screws, and that this must remain true for every value of 
x and y. 

In general, let 1} ... # 6 denote the co-ordinates of a Dyname in the first 
system, and <j> l} ... </> 6 denote those of the same Dyname in the second system. 
Let/(0!, ... 6 ) denote any homogeneous function of the first Dyname, and let 
JP (</>!, ... < 6 ) be the same function transformed to the other screws of refer 
ence. Then we have 



as an identical equation which must be satisfied whenever the Dyname de 
fined by lt ... 6 is the same as that defined by < u ... </> 6 . We must there 
fore have 

f(xa.i + yfi-i , . . . aJOe + y/3 6 ) = 



These expressions being homogeneous, they may each be developed in 

ascending powers of - . But as the identity must subsist for every value of 
oc 

this ratio, we must have the coefficients of the various powers equal on both 
sides. The expression of this identity gives us a series of equations which 



are all included in the form* 

d d 





* 



* See Proceedings Roy. Irish Acud., Ser. n. Vol. in. ; Science, p. 601 (1882). 

18-2 



276 THE THEORY OF SCREWS. [261- 

The functions thus arising are well known as " emanants " in the theory 
of modern algebra. The cases which we shall consider are those of n = 1 and 
n = 2. In the former case the emanant may be written 

- df n df 



262. Angle between Two Screws. 

It will of course be understood that/ is perfectly arbitrary, but results of 
interest may be most reasonably anticipated when / has been chosen with 
special relevancy to the Dyname itself, as distinguished from the influence 
due merely to the screws of reference. We shall first take for / the square 
of the intensity of the Dyname, the expression for which is found ( 35) 
to be 



where (12) denotes the cosine of the angle between the first and second 
screws of reference, which are here taken to be perfectly arbitrary. The 
second group of reference screws we shall take in a special form. They are 
to be a canonical co-reciprocal system, so that 

R = (\ + X 2 ) 2 + (\ 3 + X 4 ) 2 + (X 5 + X 6 ) 2 . 
Introducing these values, we have, as the first emanant, 



X 2 ) + (p a 4- /i 4 ) (X 3 + X 4 ) + (fjL 6 + fi t ) (X 5 + X 6 ) ; 



but in the latter form the expression obviously denotes the cosine of the 
angle between a and /3 where the intensities are both unity ; hence, whatever 
be the screws of reference, we must have for the cosine of the angle between 
the two screws the result 



263. Screws at Right Angles. 

In general we have the following formula for the cosine of the angle 
between two Dynames multiplied into the product of their intensities : 

dR n dR dR 



This expression, equated to zero, gives the condition that the two Dynames 
be rectangular. 



265] 



EMANANTS AND PITCH INVARIANTS. 



277 



If three screws, a, @, y, are all parallel to the same plane, and if 9 be a 
screw normal to that plane, then we must have 

dR dR 



dR 
dfr 
dR 



dR 



264. Conditions that Three Screws shall be parallel to a 
Plane. 

^ * 

Since a screw of a three-system can be drawn parallel to any direction, 
it will be possible to make any three of the quantities 6 l , ... 6 6 equal to zero. 
Hence, we have as the condition that the three screws, a, /3, 7 shall be 
all parallel to a plane the evanescence of all the determinants of the type 



dR 
dR 

dR 

d 7l > 



dR 



dR 



dR 



dR 



dR 



dR 



265. Screws on the same Axis. 

The locus of the screws d perpendicular to a is represented by the 
equation 



If we assume that the screws of reference are co-reciprocal, then the 
equation just written can only denote all the screws reciprocal to the one 
screw whose co-ordinates are 



Pi di " PS da 6 

It is manifest that all the screws perpendicular to a given line cannot be 
reciprocal to a single screw unless the pitch of that screw be infinite, other 
wise the condition 

(p a +p e ) cos (f> d sin < = 

could not be fulfilled. We therefore see that the co-ordinates just written 
can only denote those of a screw of infinite pitch parallel to a. 



278 THE THEORY OF SCREWS. [265, 

If a? be a variable parameter, then the co-ordinates 

x dR a; dR 

Oti + -. ~ 7 i #6 "T ~i ~9 

4p, da, 4p 6 da 6 

must denote a screw of variable pitch x on the same screw as or. We are 
thus conducted to a more general form of the results previously obtained 
( 47). 

These expressions may be written 

! + COS flj , or a + COS O.J, . . . 

^ 2j? 2 

where a lf a 2 , ... are the angles which a makes with the screws of reference. 

266. A general Expression for the Virtual Coefficient. 

We may also consider that function of the co-ordinates of a Dyname 
which, being always proportional to the pitch, becomes exactly equal to the 
pitch when the intensity is equal to unity. More generally, we may define 
the function to be equal to the pitch multiplied into the square of the 
intensity, and it is easy to assign a physical meaning to this function. It 
is half the work done in a twist against a, wrench, on the same screw, where 
the amplitude of the twist is equal to the intensity of the wrench. Referred 
to any co-ordinates, we denote this function by V expressed in terms of 
Xj,... X 6 . If we express the same function by reference to six co-reciprocal 
axes with co-ordinates cfi, ..., we have the result 

p 1 a. 1 s + ...> 6 a 2 = V. 
Forming now the first emanant, we have 

2pi i A + . . . + 2p, 6 /3 6 = ^ -^- . . . + ^ 6 ; 

but the expression on the left-hand side denotes the product of the two 
intensities into double the virtual coefficient of the two screws; hence 
the right-hand member must denote the same. If, therefore, after the 
differentiations we make the intensities equal to unity, we have for the 
virtual coefficient between two screws X and yu, referred to any screws of 
reference whatever one-half the expression 

dV dV 



Suppose, for instance, that X is reciprocal to the first screw of reference, 
then 



266] EMANANTS AND PITCH INVARIANTS. 279 

This can be verified in the following manner. We have 

V = p\"\ 
dv / d 



and, therefore, if X be reciprocal to the first screw of reference, the formula 
to be proved is 



A few words will be necessary on the geometrical signification of the 
differentiation involved. Suppose a Dyname A, be referred to six co-ordinate 
screws of absolute generality, and let us suppose that one of these co 
ordinates, for instance X 1; be permitted to vary, the corresponding situation 
of X also changes, and considering each one of the co-ordinates in succession, 
we thus have six routes established along which X will travel in correspond 
ence with the growth of the appropriate co-ordinate. Each route is, of 
course, a ruled surface ; but the conception of a surface is not alone adequate 
to express the route. We must also associate a linear magnitude with each 
generator of the surface, which is to denote the pitch of the corresponding 
screw. Taking X and another screw on one of the routes, we can draw a 
cylindroid through these two screws. It will now be proved that this 
cylindroid is itself the locus in which X moves, when the co-ordinate cor 
related thereto changes its value. Let 9 be the screAV arising from an 
increase in the co-ordinate Xjj a wrench on 6 of intensity 6" has components 
of intensities #/ , . . . 6 ". A wrench on X has components X/ , . . . X 6 ". But 
from the nature of the case, 



If therefore & be suitably chosen, we can make each of these ratios 1, 
so that when 6" and X" are each resolved along the six screws of reference, 
all the components except $/ , X/ shall neutralize. But this can only be 
possible if the first reference screw lie on the cylindroid containing and X. 
Hence we deduce the result that each of the six cylindroids must pass 
through the corresponding screw of reference ; and thus we have a complete 
view of the route travelled by a screw in correspondence with the variation 
of one of its co-ordinates. 

Let the six screws of reference be 1, 2, 3, 4, 5, 6. Form the cylindroid 
(X, 1), and find that one screw 77 on this cylindroid which has with 2, 3, 4, 5, G, 
a common reciprocal ( 26). From a point draw a pencil of four rays parallel 
to four screws on the cylindroid. Let OA be parallel to one of the principal 
screws ; OX be parallel to X, Otj to 77, and Oh to the first screw of reference. 



280 THE THEORY OF SCREWS. [266, 

Let the angle AOh be denoted by A, the angle ^40?; by B, and the angle 
A OX by <f). To find the component \ we must decompose A, , a twist on 
X, into two components, one on 77, the other on the first screw of reference. 
The component on 77 can be resolved along the other five screws of reference, 
since the six form one system with a common reciprocal. If we denote by 
77 the component on 77, we then have 

X \! 77 

sin (B - A) = sin (< - B) = sin(</>-J) 

and if a and 6 be the pitches of the two principal screws on the cylindroid, 
we have for the pitch of X the equation 

p = a cos 2 </> + b sin 2 <f> ; 

also -3^- = ~ V , because the effect of a change in X, is to move the screw 
aXj d(f> ctXj 

along this cylindroid. 

Air u , sin (0 - .B) 

We have Xj = 77 -r } --- A , 

sm (0- A) 

and as the other co-ordinates are to be left unchanged, it is necessary that 
77 be constant, so that 

d\ _ ,sm(B A) 
~d$~ rj sin^(<f>-A) 

dp . . sin 2 (<4 A) 

and hence ,^- = (6 - a) sin 2<f> , . ^ - 4r . 

ctXj ^T)sm(B-A) 

A , d\ d\ dd> 

Also = ^_ = _ cos ( ( f ) _^) j 

aXj rf0 rfXj 
Hence, substituting in the equation 



we deduce a = b tan </> tan A : 

but this is the condition that X and the first screw of reference shall be 
reciprocal ( 40). 

267. Analogy to Orthogonal Transformation. 

The emanants of the second degree are represented by the equation 



when F is the function into which / becomes transformed when the co 
ordinates are changed from one set of screws of reference to another. If 
we take for / either of the functions already considered, these equations 



267] EMANANTS AND PITCH INVARIANTS. 281 

reduce to an identity ; but retaining / in its general form, we can deduce 
some results of very considerable interest. The discussion which now follows 
was suggested by the reasoning employed by Professor W. S. Burnside* in 
the theory of orthogonal transformations. 

Let us suppose that we transform the function f from one set of co- 
reciprocal screws of reference to another system. Let p 1} ... p 6 be the 
pitches of the first set, and q lt ... q 6 be those of the second set. Then we 
must have 



for each merely denotes the pitch of the Dyname multiplied into the square 
of its intensity. Multiply this equation by any arbitrary factor x and add 
it to the preceding, and we have 



d_ } f -( 2 \ 

i A.J d, \ 6 1 

Regarding &, ... /3 6 as variables, the first member of this equation 
equated to zero would denote a certain screw system of the second degree. 
If that system were "central" it would possess a certain screw to which 
the polars of all other screws would be reciprocal, and its discriminant 
would vanish ; but the screw $ being absolutely the same as p, it is plain 
that the discriminant of the second side must in such case also vanish. We 
thus see that the ratios of the coefficients of the various powers of x in the 
following well-known form of determinant must remain unchanged when 
one co-reciprocal set of screws is exchanged for another. In writing the 

d z f 
determinant we put 12 for -= , &c. 

16 1=0. 

26 

36 

46 

56 

66 + ape 

Take for instance the coefficient of of divided by that of x, which is 
easily seen to be 

1 d 2 / I d 2 f 

PI da.1 2 p 6 da s 2 

* Williamson, Differential Calculus, p. 412. 



11- 


h a;p 1 , 1 2 


, 13 


, 14 


15 


21 


, 22 + 


xp 2 , 23 


,24 


25 


31 


, 32 


, 33 + 


ps, 34 


35 


41 


, 42 


V 43 


, 44 + xp 4 , 


45 


51 


, 52 


, 53 


, 54 , 


55 


61 


, 62 


, 63 


, 64 


65 



282 



THE THEORY OF SCREWS. 



[267, 



and we learn that this expression will remain absolutely unaltered provided 
that we only change from one set of co-reciprocals to another. In this / is 
perfectly arbitrary. 

268. 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 system. 

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 ( 18). Belonging to a 
screw system of the third order we have, in like manner, three screws of 
maximum or minimum pitch, which lie along the three principal axes of 
the pitch quadric ( 173). The general question, therefore, arises, as to 
whether it is always possible to select from a screw system of the ?tth order 
a certain number of screws of maximum or minimum pitch. 

Let 1} ... # 6 be the six co-ordinates of a screw referred to n co-reciprocal 
screws belonging to the given screw system. Then the function p e , or 



is to be a maximum, while, at the same time, the co-ordinates satisfy the 

condition ( 35) 

20! s + 220 A cos (12) = 1, 

which for brevity we denote as heretofore by 



Applying the ordinary rules for maxima and minima, we deduce the six 

equations 

dR 



9 - = 

From these six equations O l , ... 9 6 can be eliminated, and we obtain the 
determinantal equation which, by writing x= 1 +po, becomes 

1 %PI, cos (21), cos (31), cos (41), cos (51), cos (61) 

cos (12), l-#p a , cos (32), cos (42), cos (52), cos (62) 

cos (13), cos (23), l-a;p 3) cos (43), cos (53), cos (63) 

cos (14), cos (24), cos (34), lasp t , cos (54), cos (64) 

cos (15), cos (25), cos (35), cos (45), lacp s , cos (65) 

cos (16), cos (26), cos (36), cos (46), cos (56), 1 - ay> 6 



= 



268] EMANANTS AND PITCH INVARIANTS. 283 

It is easily seen that this equation must reduce to the form 



In fact, seeing it expresses the solution of the problem of finding a screw 
of maximum pitch, and that the choice may be made from a system of the 
sixth order, that is to say, from all conceivable screws in the universe it is 
obvious that the equation could assume no other form. 

What we now propose to study is the manner in which the necessary 
evanescence of the several coefficients is provided for. After the equation 
has been expanded we shall suppose that each term is divided by the 
coefficient of a? that is, by 



From any point draw a pencil of rays parallel to the six screws. On 
four of these rays, 1, 2, 3, 4, we can assign four forces which equilibrate 
at the point. Let these magnitudes be X 1} X. 2 , X 3 , X 4 . We can express 
the necessary relations by resolving these four forces along each of the four 
directions successively. Hence 

X, + X, cos (12) + X s cos (13) + X, cos (14) = 0. 

X, cos (21) + X 2 +X, cos (23) + X, cos (24) - 0. 

X l cos (31) + X z cos (32) + X. + Z 4 cos (34) = 0. 

X, cos (41) + X, cos (42) + X s cos (43) + X 4 = 0. 

Eliminating the four forces we have 

1, cos (12), cos (13), cos (14) 

cos (21), 1, cos (23), cos (24) 

I cos (31), cos (32), 1, cos (34) 

cos (41), cos ^42), cos (43), 1 

Thus we learn that every determinant of this type vanishes identically. 

Had we taken live or six forces at the point it would, of course, have been 
possible in an infinite number of ways to have adjusted five or six forces to 
equilibrate. Hence it follows that the determinants analogous to that just 
written, but with five and six rows of elements respectively, are all zero. 

These theorems simplify our expansion of the original harmonic deter 
minant. In fact, it is plain that the coefficients of x*, of x, and of the 
absolute term vanish identically. The terms which remain are as follows : 

x ti + Ao? + Ex A + C ic 8 = 0. 



284 THE THEORY OF SCREWS. [268- 

, 1 
where A = 2, , 

V sin 2 (l, 2) 



in which 

1, cos (12), cos (13) 

cos (12), 1, cos (23) 
cos (13), cos (23), 1 

If by S (123) we denote the scalar of the product of three unit vectors 
along 1, 2, 3, then it is easy to show that 



We thus obtain the following three relations between the pitches and the 
angular directions of the six screws of a co-reciprocal system*, 



Pi 



The first of these formulae gives the remarkable result that, the sum of the 
reciprocals of the pitches of the six screws of a co-reciprocal system is equal 
to zero. 

The following elegant proof of the first formula 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 174 that the pitch 
quadric 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 ( 176). 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. Remembering that the 
sum of the squares of the two sets of conjugate diameters is equal, the 
required theorem is at once evident. 

* Proceedings of the Royal Irish Academy, Series in. Vol. i. p. 375 (1890). A set of six 
screws are in general determined by 30 parameters. If those screws be reciprocal 15 conditions 
must be fulfilled. The above are three of the conditions, see also 271. 



270] EMANANTS AND PITCH INVARIANTS. 285 

269. 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 system 
of the nth order, is a constant for each screw system. 

Let A be the given screw system, and B the reciprocal screw system, 
Take 6 n co-reciprocal screws on B, and any n co-reciprocal screws 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 considerations, that the 
sum of the reciprocals of the pitches of co-reciprocals is constant for the 
screw system of the second and third order ( 40, 176), 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 system is evidently 
a characteristic feature of that screw system. 

270. Theorem as to Signs. 

If in one set of co-reciprocal screws of an n-system there be k screws with 
negative pitch and n k screws with positive pitch, then in every set of 
co-reciprocal screws of the same system there will also be k screws with negative 
pitch and n k screws with positive pitch. 

To prove this we may take the case of a five-system, and suppose that of 
five co-reciprocals A l , A 2 , A 3 , A 4 , A 5 the pitches of three are positive, say 
mf, ra 2 2 , TO/, while the pitches of the two others are negative, say ra 4 2 , 



Let 8 be any screw of the system, then if 1} ... # 5 be its co-ordinates 
with respect to the five co-reciprocals just considered, we have for the pitch 
of 6 the expression 

raM 2 + w 2 2 2 2 + ra 3 2 <9 3 2 - m 4 2 4 2 - m 5 2 5 2 . 

Let us now take another set of five co-reciprocals B 1} B 2 , B 3 , B 4 , B 5 
belonging to the same system, then the pitches of three of these screws must 
be + and the pitches of two must be . For suppose this was not so, but 
that the five pitches were, let us say n^, n^, w 3 2 , v? 4 2 , w s 2 . Let the co-ordinates 
of with respect to these new screws of reference be <f> 1} <f> 2 , ... B , then 
the pitch will be 

V& 2 + n.*<f> 2 2 + n s a 3 2 + nty* - n 5 2 </> 5 2 . 



286 THE THEORY OF SCREWS. [270- 

Equating these two values of the pitch we ought to have for every screw S 



- n^ + M 3 2 < 2 2 + w 3 2 < 3 2 -f n 4 2 < 4 2 + m* 
But it can easily be seen that this equation is impossible. 

Let H be the screw to which all the screws of the five-system are re 
ciprocal, and let us choose for 8 the screw reciprocal to A 1} A 2 , A 3 , B 6 , H. 
The fact that S is reciprocal to H is of course implied in the assumption that 
8 belongs to the five-system, while the fact that S is reciprocal to each of the 
screws A lt A 2 . A 3 , B 5 gives us 

1= =0, 2 = 0, 3 = 0, 8 = 0. 
Hence we would have the equation 



which would require that all the co-ordinates were zero, which is im 
possible. 

In like manner any other supposition inconsistent with the theorem of 
this article would be shown to lead to an absurdity. The theorem is there 
fore proved. 

We can hence easily deduce the important theorem that three of the 
screws in a complete co-reciprocal system of six must have positive pitch 
and three must have negative pitch*. 

For in the canonical system of co-reciprocals the pitches are + a, a, 
4- b, b, + c, c, i.e. three are positive and three are negative, and as in this 
case the w-system being the six -system includes every screw in space we see 
that of any six co -reciprocals three of the pitches must be positive and three 
must be negative. 

271. Identical Formulae in a Co-reciprocal System. 

Let any screw a be inclined at angles cxi, a2, ... aG to the respective six 
screws of a co-reciprocal system. 

Then we have for the co-ordinate a n 

_ (p a +p n )cos al d al sin al 
^ ~~~ 



* This interesting theorem was communicated to me by Klein, who had proved it as a 
property of the parameters of "six fundamental complexes in involution" (Math. Ann. Band. 
i. p. 204). 



273] EM AN ANTS AND PITCH INVARIANTS. 287 

If we substitute these values for a l , ... a n in the expression 



we obtain the equation 

cos 2 al cos 2 a2 cos 2 a6~ 

+ - + ...+ - 

Pi P* P 



f 

=/>a 



cos a6 (p 6 cos a6 d a6 sin a6) 

_|_ __ _ . ^ 

JP 

( p l cos a l d al sin al) 2 ( p 6 cos a6 d a6 sin a6) 2 

As al, &c., da, &c., >!, &c. are independent of p a we must have the three 
co-efficients of this quadratic in p a severally equal to zero. 

272. Three Pitches Positive and Three Negative. 

The equation 

cos 2 al cos 2 a2 cos 2 a6 _ 

Pi P-2 p% 

also shows that three pitches of a set of six co-reciprocals must be positive 
and three must be negative. For, suppose that the pitches of four of the 
co-reciprocals had the same sign, and let a be a screw perpendicular to the 
two remaining co-reciprocals, then the identity just written would reduce to 
the sum of four positive terms equal to zero. 

From this formula and also 

11 1 

--+- + ... + -=0 

Pl P-2 P6 

sin 2 ai sin 2 a2 sin 2 6 
we have h + . . . H = 

Pi PI PS 

273. Linear Pitch Invariant Functions. 

We propose to investigate the linear functions of the six co-ordinates of 
a screw which possess the property that they remain unaltered notwith 
standing an alteration in the pitch of the screw which the co-ordinates 
denote. It will first be convenient to demonstrate a general theorem which 
introduces a property of the six screws of a co-reciprocal system. 

The virtual coefficient of two screws is, as we know, represented by half 
the expression 

(p a + pp) cos 6 d sin 6, 

where p a and p ft are the pitches, is the angle between the two screws, 
and d the shortest perpendicular distance. The pitches only enter into 



288 THE THEORY OF SCREWS. [273, 

this expression by their sum; and, consequently, if p a be changed into 
p a + x, and pp be changed into p$ x, the virtual coefficient will remain 
unaltered whatever x may be. 

We have found, however ( 37), that the virtual coefficient admits of 
representation in the form 

Pii/3i+ ... + profit- 

To augment the pitch of a by x, we substitute for a l} a..,, ... the several 
values ( 265), 

SY> fit* 

a, + - cos a,!, a 2 + -^ cos a 2 , ... 
2^] 2p 2 

where a 1( a 2 , ... are the angles made by the screw a with the screws of 
reference. Similarly, to diminish the pitch of ft by x, we substitute for 
&, /3 2 , . the several values 

OC OG 

&-H-COS&!, &-a- cos& 2 , &c. 

Zpi Zp 2 

With this change the virtual coefficient, as above expressed, becomes 

x \ ( Q x 

+ 9~ COS a - COS 

\ ^Pi 

or, 

CO 



o v/ ^j COS ttj + p 2 COS a 2 + ...! COS Oj 2 COS 2 . . . ) 

Z 

s ! cos &! cos a 2 cos & 2 cos a e cos 6, 

We have already shown that such a change must be void of effect upon 
the virtual coefficient for all values of x. It therefore follows that the 
coefficients of both x and x* in the expressions just written must be zero. 
Hence we obtain the two following properties : 

= (p\ cos j + ... + /3 6 cos a 6 ) (^ cos h+ ... + a 6 cos 6 6 ), 
_ cos a : cos bi cos a e cos 6 6 

The second of the two formulas is the important one for our present 
purpose. It will be noted that though the two screws, a and @, are com 
pletely arbitrary, yet the six direction cosines of a. with regard to the screws 
of reference, and the six direction cosines of /3 with regard to the same 
screws of reference, must be connected by this relation. Of course the 
equation in this form is only true when the six screws of reference are 
co-reciprocal. In the more general case the equivalent identity would be 
of a much more complicated type. 



274] EMANANTS AND PITCH INVARIANTS. 289 

274. A Pitch Invariant. 

Let h lt ... h 6 be the direction angles of any ray whatever with regard to 
six co-reciprocal screws of reference, then the function 

U = ! cos h-i + . . . + a s cos h s 
is a pitch invariant. 

For, if we augment the pitch of a by x, we have to write for a lf ... a 6 
the expressions 

x x 

j + 5 COS ! . . . 6 + COS 6 , 

API Ap Q 

and then U becomes 

! COS A! + . . . + COS h K 

x /cos a a cos ^ cos a cos A, 

~r o I T H -- 



Pi 

but from what we have just proved, the coefficient of x is zero, and hence 
we see that 

! cos ^ -|- . . . + 6 cos h 6 

remains unchanged by any alteration in the pitch of a. 

If we take three mutually rectangular screws, a, /3, 7, then we have the 
three pitch invariants 

L = 1 cos ^ + . . . + 6 cos a 6 , 
M = #! cos &J + ... + # 6 cos 6 6 , 

N = O l COS Ci + . . . + # 6 COS C 6 . 

It is obvious that any linear function of L, M, N, such as 

fL + gM+hN, 
is a pitch invariant. 

We can further show that this is the most general type of linear pitch 
invariant. 

For the conditions under which the general linear function 

^0,+ ... +A n n 
shall be a pitch invariant are that equations of the type 

A l cos a, A 6 cos a a _ e 

-- r . . . + - = U ; dec. 

Pi P 

shall be satisfied for all possible rays. 

Though these equations are infinite in number, yet they are only equi 
valent to three independent equations ; in other words, if these equations 
are satisfied for three rays, a, b, c, which, for convenience, we may take to 
be rectangular, then they are satisfied for every ray. 

B. 19 



290 THE THEORY OF SCREWS. [274- 

For, take a ray e, which makes direction-angles X, /A, v with a, b, c, then 
we have 

COS j = COS X COS ! + COS /A COS &! + COS V COS Cj , 



cos e 6 = cos X cos a 6 + cos /u, cos 6 6 + cos v cos c 6 . 
Hence 





A 6 cos e 6 ^ A 


! COS 0^ ( ^^ ^.^IjCOSO! 


Pi 




Pi Pi 












+ r>na 11 > 








Pi 


If, therefore, we 


have 




r 


4i cos i ^ A l cos I 


i . v^i cosc i_ 


, *w 








Pi 



then, for every ray, we shall have 

cos 



ft 

It thus follows that the coefficients of a linear function which possesses 
the property of a pitch invariant must be subjected to three conditions. 
There are accordingly only three coefficients left disposable in the most 
general type of linear pitch invariant. Now, 

fL+gM + hN 

is a pitch invariant which contains three disposable quantities, f, g, h; it 
therefore represents the most general form of linear function which possesses 
the required property. 

We have thus solved the problem of finding a perfectly general expression 
for the linear pitch invariant function of the co-ordinates of a screw. 

It is convenient to take the three fundamental rays as mutually rect 
angular; but it is, of course, easy to show that any linear pitch invariant 
can be expressed in terms of three pitch invariants unless their determining 
rays are coplanar. We may express the result thus : Let L, M, N, be 
four linear pitch invariants, no three of which have coplanar determining 
rays. Then it is always possible to find four parameters, X, /*, v, p, such that 
the following equation shall be satisfied identically : 

\L + fiM + vN + pO = 0. 



275. Geometrical meaning. 

The nature of the pitch invariant function can be otherwise seen. It is 
well known that in the composition of two or more twist velocities we 



276] EMANANTS AND PITCH INVARIANTS. 291 

discover the direction of the resultant screw and the magnitude of the 
resultant twist velocity by proceeding as if the twist velocities were vectors. 
Neither the pitches of the component screws nor their situations affect the 
magnitude of the resultant twist velocity or the direction of the resultant 
screw. This principle is, of course, an immediate consequence of the law 
of composition of twist velocities by the cylindroid. 

Let any ray a make an angle A, with the ray 6, and angles a 1} ... a 6 with 
the six screws of reference. The twist velocity $ on 6 if resolved on a has 
a component 0cos\. This must be equal to the sum of the several com 
ponents 1} # 2 , ... resolved on a; whence we have 

cos X = #! cos a-i + ... + G cos a 6 . 
If we make = unity, we obtain 

cos \=0 1 cos a t + ... +0 6 cos a s . 

This gives a geometrical meaning to the pitch invariant. It is simply the 
cosine of the angle between the screws 6 and a. As, of course, the pitch 
is not involved in the notion of this angle, it is, indeed, obvious that the 
expression for any function of the angle must be a pitch invariant. 

We now see the meaning of the equation obtained by equating the pitch 
invariant to zero. If we make 

1 COS !+...+ # 6 COS C1 6 = 

it follows that a. and must be at right angles. The equation therefore 
signifies the locus of all the screws that are at right angles to a. 

The two equations 

1 cos !+... + 6 cos a 6 = 0, 

0! cos &!+... -f # 6 cos b 6 = 0, 

denote the screws perpendicular to the two directions of a and /3. In other 
words, these two equations define all the screws perpendicular to a given 
plane. 

276. Screws at infinity. 

Let us now take the case where a, ft, 7 are three rectangular screws, and 
examine the conditions satisfied by 1} ..., 6 when subjected to the three 
following equations : 

#1 cos (&!+ ... + 6 cos a 6 = 0, 

0! COS h + ... + 6 COS b 6 = 0, 
l COS G! + ... + 6 COS C 6 = 0. 

The screws which satisfy these conditions must all be perpendicular to the 

192 



292 THE THEORY OF SCREWS. [276, 

three directions of o, j3, 7. For real and finite rays this is impossible ; for 
real and finite rays could not be perpendicular to each of three rays which 
were themselves mutually rectangular. This is only possible if the rays 
denoted by 1} ... 6 K are lines at infinity. 

It follows that the three equations, L = 0; M=Q; N=Q, obtained by 
equating the three fundamental pitch invariants to zero, must in general 
express the collection of screws that are situated in the plane at infinity. 

We can write the three equations in an equivalent form by the six 
equations 

., ,,COSi COS&j 7 COSCi 

0i =/ + g- -+A- -, 

Pi Pi Pi 



n , cos a s cos 6 6 7 cos c 6 
t/6 =/ - + g + fi - , 
Pa p 6 p s 

where f, g, h are any quantities whatever; for it is obvious that, by substi 
tuting these values for 1 , ... 6 in either L, or M, or N, these quantities are 
made to vanish by the formula of the type 

X cos a 6 cos 6 



Pi 

We have, consequently, in the expressions just written for B lt ... 6 , the 
values of the co-ordinates of a screw which lies entirely in the plane at 
infinity. 

277. Expression for the Pitch. 

It is known that if a, 0, 7 be the direction-angles of a ray, and if P, Q, R 
be its shortest perpendicular distances from three rectangular axes, then 

P sin a cos a + Q sin /3cosfi + R sin 7 cos 7 = 0. 

Let 77, , be three screws of zero pitch, which intersect at right angles, and 
let 6 be another screw, then, if vr^ be the virtual coefficients of 77 and 0, 

2"5T,,0 = p 6 cos a P sin a, 
whence, by the theorem just mentioned, we have 

p e = Zvr^o cos a + 2tjy| e cos + 2^ cos 7. 

Let a 1} ... a 6 be the angles made by 77 with the six co-reciprocal screws of 
reference, then 

cos a = L = 6 1 cos a x + . . . + 6 cos a a , 
and, similarly, for the two other angles, 

cos yS = M= e i cos &J + ...+ 6 cos 6 6 , 
cos 7 = N = l cos d + . . . + B 6 cos c 6 , 



277] EM AN ANTS AND PITCH INVARIANTS. 293 

and L 2 + M 2 -\- N*= 1, whence we have for the pitch the homogeneous ex 
pression 



It appears from this that the three equations, 

57,0 = ; -GTf fl = ; Tf fl = 0, 

indicate that 6 must be one of a pencil of rays of zero pitch radiating from 
a point. 

The equations Z = ; lf=0; N=Q, define a screw of indeterminate 
pitch. 

Why the screws in the plane at infinity ( 46) should in general present 
themselves with indeterminate pitch is a point which requires some ex 
planation. The twist about such a screw, as around any other, consists, of 
course, of a rotation and a translation. If, however, the finite parts of the 
body are only to be moved through a finite distance, the amplitude of the 
twist must be infinitely small, for a finite rotation around an axis at infinity 
would, of course, imply an infinitely great displacement of parts of the body 
which were at finite distances. The amplitude of the rotation is therefore 
infinitely small, so that, if the pitch is finite, the displacement parallel to 
the axis of the screw is infinitely small also. It thus appears that the effect 
of a small twist about a screw of any finite pitch at infinity is to give the 
finite parts of the body two displacements, one of which is infinitely insig 
nificant as regards the other. We can therefore overlook the displacement 
due to the pitch, and consequently the pitch of the screw unless infinite is 
immaterial ; in other words, in so far as the screw is the subject of our 
investigation, its pitch is indeterminate. 

In like manner we can prove that a screw in the plane at infinity, when 
regarded as the seat of a wrench, must, when finite forces are considered, 
be regarded as possessing an indeterminate pitch. For, let the force apper 
taining to the wrench be of finite magnitude, then its effect on bodies at 
finite distances would involve a couple of infinite moment. It therefore 
follows that the force on the screw at infinity must be infinitely small if the 
effects of the wrench are finite. The moment of the couple on the screw 
of finite pitch is therefore infinitely small, nor is its magnitude increased 
by importation from infinity ; therefore, at finite distances, the effect of the 
couple part of the wrench may be neglected in comparison with that of the 
force part of the wrench. But the pitch of the screw is only involved so 
far as the couple is concerned ; and hence whatever be the pitch of the 
screw lying in the plane at infinity, its effect is inoperative so far as finite 
operations are concerned. 



294 THE THEORY OF SCREWS. [277, 

There is here a phenomenon of duality which, though full of significance 
in non-Euclidian space, merely retains a shred of its importance in the space 
of ordinary conventions. A displacement, such as we have been considering, 
may of course arise either from a twist about a screw of infinite pitch at an 
indefinite distance, or a twist about a screw of indefinite pitch at an infinite 
distance. 

278. A System of Emanants which are Pitch Invariants*. 

From the formula 

2tzr a/3 = (p a + pp) cos (a/3) - d a $ sin (a/3), 



we obtain 



sin () = i (p a + p ft ) (a, ^ + . . . + 6 ^-J 



-I* ,-+...+* -}ft^}\ 
d/3, dftJ V JR. r 



or from symmetry 



We thus obtain an emanant function of the co-ordinates of a and /3 which 
expresses the product of the shortest distance between a and /8 into the sine 
of the angle between them. The evanescence of this emanant is of course 
the condition ( 228) that a and /3 intersect. 

This emanant is obviously a pitch invariant for each of the two screws 
involved. It will be a pitch invariant for a whatever be the screw /3. Let 
us take for /3 the first screw of reference so that 

& = 1; & = 0... /3 6 = 0. 

Then 

A (P*+PI\ 

da, \ \/R a ) 
must be a pitch invariant. It may be written 



2 da, 
This article is due to Mi- A. Y. G. Campbell. 



278] EM AN ANTS AND PITCH INVARIANTS. 295 

but we know that the last term is itself a pitch invariant, and hence we have 
the following result. 

If Pa be the pitch of a screw a expressed in terms of the co-ordinates, and 
if R a denote the function a + ... + 2 + 2^0, cos (12) + ... = 1, then the 
several functions 

&_ _Pa 

da, 



remain unaltered if instead of j . . . a the co-ordinates of any other screw on 
the same straight line as a should be substituted. 

This is easily verified by the known formulae that if a be any screw and 
another screw on the same axis as a whose pitch is p a + ac, then 

x dR Q _ x dR 

01 = ai + ^ 1 d^ i " 6 - a6 + 4p 6 da 6 
whence 

- s - 2 + * ~ * + A> cos al 



CHAPTER XXL 

DEVELOPMENTS OF THE DYNAMICAL THEORY. 

279. Expression for the Kinetic Energy. 

Let us suppose that a body of mass M is twisting around a screw a with 
the twist velocity a. It is obvious that the kinetic energy of the body must 
be the product of Md 2 and some expression which has the dimensions of the 
square of a linear magnitude. This expression has a particular geometrical 
significance in the Theory of Screws, and the symbols of the theory afford a 
representation of the expression in an extremely concise manner. 

Let 77 be the impulsive screw which corresponds to a as an instantaneous 
screw, the body being supposed to be perfectly unconstrained. 

As usual p a is the pitch of a and (a?/) is the angle between a and 77. 
From the formulae of 80 we have, where H is a common factor, 

H^ = + aa^; Hv]2 = a 2 ; 

#773 = + ba 3 ; Hr) 4 = - 6a 4 ; 

Hr) 5 = + ca & ; #77,3 = -ca 6 ; 
whence 

H [(771 + 7? 2 ) (! + 2 ) + (r; 3 + 774) (a 3 + a 4 ) + (775 + 77 6 ) (a fl + a e )] 

= a (! 2 - a/) + b (a 3 2 - a/) + c (a 5 2 - a 6 2 ) =p a 
and we obtain 



cos 
The kinetic energy is 



in 

" 



cos (an) 



cos 
* Trans. Roy. Irish Acad., Vol. xxxi. p. 99 (1896). 



279, 280] DEVELOPMENTS OF THE DYNAMICAL THEORY. 297 

which is the required expression for the kinetic energy. It is remarkable 
that the co-ordinates of the rigid body are introduced by the medium of the 
impulsive screw alone. 

280. Expression for the Twist Velocity. 

If an impulsive wrench of unit intensity on a screw 77 be applied to a 
quiescent rigid body of unit mass which in consequence commences to twist 
about an instantaneous screw a, it is required to find the initial twist 
velocity a. 

The impulsive wrench may be replaced by component impulsive forces 
T]I, ,.. rj 6 on the six principal screws of inertia and component impulsive 
couples with moments 0%, atj^, br} 3 , 6774, crj 5 , cij s about those screws 
of inertia. 

The force ^ is expressed by the velocity it produces in the unit mass 
parallel to the direction of 77. The component twist velocity of a is di 
about the first principal screw, and accordingly the velocity of translation 
parallel to that screw is adaj. Hence we have 



but ! = ~-. T?I, 

cos (a?;) 

whence we obtain* 

cos (OCT?) 

f* . . _ v / 

P 

It will be noted that in this expression the co-ordinates of the rigid body 
are introduced through the medium of the impulsive screw alone. 

A special case arises when the impulsive wrench is a couple, in which 
case of course p^= oo . As the effect of an impulsive couple is to produce 
a pure rotation only, we must under these circumstances have p a = 0. 
Poinsot s well-known construction exhibits the axis of the initial rotation 
as the diameter of the mornental ellipsoid conjugate to the plane of the 
impulsive couple. As three conjugate diameters of an ellipsoid could not 
lie in the same plane, it follows that in the case of p^ = oo we can never 
have a and 77 at right angles. As p a is zero while cos (777) is not zero, we 
must have d infinite. 

This might have been inferred from the fact that as the intensity of the 
impulsive wrench was not zero while the pitch of the screw on which it lay 
was infinite, the moment of the impulsive couple was infinite and conse 
quently the initial twist velocity must be infinite. 

* Trans. Roy. Irish Acad., Vol. xxxi. p. 100 (1896). 



298 THE THEORY OF SCREWS. [280- 

Unless in this exceptional case where p^ is infinite it is always true that 
when p a is zero, a and 77 are at right angles. 

It is universally true that when the impulsive screw and the instan 
taneous screw are at right angles (the body being quite free), the pitch of 
the instantaneous screw must be zero. 

For if p a were not zero when cos (a?;) was zero then a. must be zero. As 
some motion must result from the impulse (the mass of the body being 
finite) we must have p a infinite. The initial motion is thus a translation. 
Therefore the impulse must have been merely a force through the centre 
of gravity ; a and 77 must be parallel and cos (a?;) could not be zero. 

The expression for the kinetic energy in 279, 



cos 



assumes an indeterminate form when the impulsive wrench reduces to a 
couple. For we then have p a = 0, but as cos (a?;) is not zero the expression 
for v? ari , i.e. 

2 {(POL + pj cos (CM?) - d ar , sin (arj)}, 
becomes infinite. 

The expression for the kinetic energy arising from an impulsive wrench 
of unit intensity on a screw 77 applied to a free body of unit mass which 
thereupon begins to twist with an instantaneous movement about a screw a 
has the concise form 



Po 

281. Conditions to be fulfilled by two pairs of Impulsive and 
Instantaneous Screws. 

Let a be a screw about which a free rigid body is made to twist in 
consequence of an impulsive wrench administered on some other screw, 77. 
Let /3 be another instantaneous screw corresponding in like manner to as 
an impulsive screw. Then we have to prove that the two following formulas 
are satisfied * : 

cos^) cos W + ~To^ cos () = 2l 

Pa 



/ \ w DT) / / i <- \ ** 

cos (ttq) cos (pt) 

* Proceedings of the Camb. Phil. Soc., Vol. ix. Part iii. p. 193. 



282] DEVELOPMENTS OF THE DYNAMICAL THEORY. 299 

To demonstrate the first of these formulae. Expand the left-hand side and 
it becomes 



ft + &) (17, + 

i + ,) (ft + ft) + (a + ,) (ft + ft) + ( B + a.) (ft + ft)}. 



But, as already shown, 



cos (a?;) cos (a?;) 



whence, by substitution, the expression reduces to 

+ a (A + A) (i - 2 ) + a (! + a.,) (A - /3 2 ) 
+ 6 (A + ft) (. - 4 ) + 6 (a. + a 4 ) (/8, - /3 4 ) 
+ c (/3 6 + A) ( 5 - a e ) + c ( B + 6 ) (/3 5 - A) 

4 + 2ca 5 /? 5 - 



To prove the second formula it is only necessary to note that each side 
reduces to 



It will be observed that these two theorems are quite independent of the 
particular screws of reference which have been chosen. 

282. Conjugate Screws of Inertia. 

We have already made much use of the important principle that is 
implied in the existence of conjugate screws of inertia. If a be reciprocal 
to then must 17 be reciprocal to . This theorem implied the existence 
of some formula connecting -nr a f and CT^. We see this formula to be 

Pa Pft 



cos ewj cos 

We have now to show that if ^^ = 0, then must w a f = 0. 

Let us endeavour to satisfy this equation when tn-^ is zero otherwise 
than by making uj af zero. Let us make p a infinite, then -n^ will reduce 
to %p a cos (af ) (for we may exclude the case in which p ( is also infinite 
because in that case r af = 0, inasmuch as any two screws of infinite pitch 
are necessarily reciprocal). 



300 THE THEORY OF SCREWS. [282, 

The formula becomes, when p a is very large, 

Pa- P? 1 / *-\ 

*-r r ura = - ~~ir. I p a cos (a|). 
cos(cc?7) cos(/3|) " 

In this case as the twist about a is merely a translation, we must have 
cos err) 1, so that 



is to vanish, but this cannot be secured by making p? zero, because that 
cannot happen without cos (/3) being zero, except the pitch of be infinite 
( 280) which is the case already excluded. It is therefore necessary that 
cos (a) be zero, but this requires that a and be reciprocal, i.e. that vr a = 0. 

Let us now suppose that we try to satisfy the original equations by 
making OT^ = 0, pp = 0. Here again we find that Pfs = entails cos (/3) zero, 
except p$ = GO . This in general makes -CT^ infinite so that the equation is 
not satisfied. If a and were at right angles then no doubt the equation 
would be satisfied, but then -S7 a | is zero. We thus see that notwithstanding 
the special form of the fundamental equation ( 281) it implies no departure 
from the complete generality of the principle that whenever is^ is zero 
then must <r f be also zero. 

283. A Fundamental Theorem. 

Let us suppose that a rigid body is either entirely free or constrained in 
any manner whatever. Let r) be an impulsive screw whose pitch p^ is not 
infinite. Let 77 " be the intensity of an impulsive wrench on that screw, it 
being understood that ?/" is to be neither zero nor infinity. Let a be the 
instantaneous screw about which the body, having been previously quiescent, 
will commence to twist with an instantaneous twist velocity a. It is also 
supposed that p a is neither zero nor infinity. 

Let be the impulsive screw similarly related to /9, and let the affiliated 
symbols have the corresponding significations and limitations. 

Let be the impulsive screw similarly related to 7, and let the affiliated 
symbols have the corresponding significations and limitations. 

The instantaneous movement of the body must necessarily be the same 
as if it had been quite free and had received in addition to the impulsive 
wrench of intensity 77 " on the screw rj, an impulsive wrench of intensity p" 
situated on some screw p belonging to the system of screws reciprocal to the 
freedom of the body. 

Let these two wrenches compound into a single wrench of intensity to " 
on a screw w. 



283] DEVELOPMENTS OF THE DYNAMICAL THEORY. 301 

Then we have ( 279), 



, cos (aco) 

Ct = CO - 



and also ( 278), 



Pi<*i f x p 2 a-2 x p (i a 6 f , 

&>! = cos (&)), eo 2 = ~ cos (CLCO), . . . o> 6 = cos (aw). 

Pa Pa Pa 

But from the fact that to " is the resultant of ?/" and p" we must have by 
resolving along the screws of reference 

/ / / / /// /// /// /// / / /// // /.\ 

CO C0 1 = T) 77J + P PI, CO C0 2 =T) 1J 2 + p p 2 , . . . CO CO = r) T] ti + p p 6 ...... (1), 

whence we obtain by substitution, 

>!! = W + p "pi, dp. 2 CL 2 = if" I* + p"p2, djOflOe = rj "r} 6 + p "p K . . .(ii). 

If we multiply the first of these equations by^/Su the second by^ 2 /3 2 , &c., 
and then add, we obtain 



as however p is on the reciprocal system we must have, except when p" = <x> 
to be subsequently considered, 

d^p^a^^r) "^^. 
In like manner, 

d2p 1 2 a 1 7i = >/" C3 Vr 
We shall similarly find 

ivi = r"^> faptfi* = ?"* } 

} ............... (Hi), 

l 1 = r^> 7^ 2 7i#i = ?""K I 

whence by multiplication 



But we have chosen the intensities rj ", % ", % " so that no one of them is 
either zero or infinity, whence 

T\P Sy ^ = OT iy OT far^ ........................... (iv). 

It remains to see whether this formula will continue to be satisfied in the 
cases excepted from this demonstration. 

Let us take the case in which p^ is infinite, which makes ^ ... infinite. 
We have in the case of p^ very large, 

ffr, cos (771) 
* = ~2p^> 
the equations (ii) become 

TJ"p^ cos (771 ) . T; "^,, cos (776) 

= Z = 



302 THE THEORY OF SCREWS. [283, 

multiplying the equations severally by pi,2h> an d adding, we get 
dSpiXA = W Pi (Pi cos ("ni ) + & cos (772 )+...+ &j cos (176)) 



= ^"TSfa (since p^ is indefinitely large), 

whence we proceed as before and we see that the theorem (iii) remains true, 
even if p^ or p$ or pg be infinite. 

If p a be zero, then in general cos aeo is zero. But in this case p a -f- cos aw 
becomes d a the length of the perpendicular from the centre of gravity upon 
a. Hence we have 



and the proof proceeds as before so that in this case also the theorem holds 
good. 

Finally, let jj a be infinite, <u must then be of zero pitch and pass through 
the centre of gravity and 

dp a = CD ". 
We have 



o> 1 = cosai, o) 3 = cos 
so that the equations (i) become 

| a>" cos (ai ) = 77" rj l + p "pi , \u>" cos (ai ) = r) "t] 2 + p "p 2 , 
| w" cos (as ) = 7/"7? 3 + p^Vs, 2 <u/ " cos ( a 3 ) = 7y" 74 + p "pi, 
\<a>" COS (as) = r) "<r)- a + p "p 5 , \<o" COS (05) = ^ "^ + p"p s . 



Multiplying these equations by +a^ lt a/3 2 , +b/3 3 , 6/3 3 , ... and adding, 
we have 

| <o" [a (fa - /8 a ) cos (ai ) + b (j3 3 - /3 4 ) cos (as ) + c (/8 5 - /S 6 ) cos (as )] = 17 "^ 

Let a- be the screw belonging to the reciprocal system on which there is 
an impulsive wrench of intensity a " due to the reactions when an impulsive 
wrench is administered on . Then we have 

/3a/3i = ^ "^i + o-" o-j ; - $afi t = %" % z + o-"V 2 , 
whence 

/8a (fii y8 2 ) cos (ai ) = f " cos (|i ) cos (ai ) + a" cos (<ri ) cos (ai ), 
with similar expressions for the two other pairs, whence by addition 

/8 {a (fr - yS 2 ) cos (ai )+ b (/3 3 - /9 4 ) cos (as) + c (J3 S - yS 6 ) cos (as)] = %" cos (a|), 
for since a and a- are reciprocal and p a = oo we must have cos (a<r) = 0. 



284] DEVELOPMENTS OF THE DYNAMICAL THEORY. 303 

We thus obtain 

1 / / 

and similarly 



whence 7 cos()_ tsr,, 

wiieuut; - ^ -- - -- = - 

COS (0 OT^ y 

or remembering that p a is infinite, 

j-/// 

2L CTg * _ OT i0 

fa " a V,y 

But we had already from (iii), 



whence we deduce that in this case also the formula remains true. We thus 
obtain the following general theorem. 

If 1), , be three impulsive screws and a, /3, 7 the three corresponding 
instantaneous screws, then in all cases, no matter how the movements of the body 
may be limited by constraints, the following formula holds good*: 



It is easily shown that this relation subsists when the correspondence 
between ?/ and a is of the more general type implied by the equations 

dU 



where U is any homogeneous function of the second degree in the co 
ordinates. 

284. Case of a Constrained Rigid Body. 

Let 77 and be, as before, a pair of impulsive screws, and let a and ft 
be the corresponding pair of instantaneous screws. Let p be the screw on 
which a reaction is contributed by the constraints at the moment when the 
impulsive wrench is applied on 77. 

The movement of the body twisting about a is therefore the same as if 
it had been free, and one impulsive wrench had been imparted about 77 
and another simultaneously about p, so that the following conditions are 
satisfied : 



* Trans. Eoy. Irish Acad., Vol. xxx. p. 575 (1894). 



304 THE THEORY OF SCREWS. [284- 

Multiplying by p^, ...p (i a , respectively, and adding, we have 

<*Uaa = ^ "tff^ , 

where u^ = pftf + . . . + p<? <*<? ; 

because, as p belongs to the reciprocal system, we must have 

OT O = 0. 



Similarly, if we multiplied the six equations by p^i, ...p 6 /3 6 , respectively, 
and added, we should get, since p is reciprocal to /3 also, 



where u aft = p^a^ + . . . + p?a t . 

Eliminating d and rj " we have the concise result*, 

Uaa. &pT, ~ Ua^ar,. 

In a similar way we can deal with the pair of screws, /3 and , and by 
eliminating <r, the reaction of the constraints in this case, we obtain the 
result 



Finally, from these two equations we can eliminate w a/3 , and we obtain 



This formula is a perfectly general relation, connecting any two pairs of 
impulsive and instantaneous screws 77, a and , /3. It holds whether the 
body be free or constrained in any way whatever. If the body be perfectly 
free, then it is easy to show that it reduces to the result already found, viz. 

P* PP 

ooBfo) * cos(/3) 

285. Another Proof. 

From the theory of impulsive and instantaneous screws in an n-system 
we know (97) that if a lf ... ct n be the co-ordinates of an instantaneous screw, 
then the co-ordinates i) 1} ... 17 w of the reduced impulsive screw may be deter 
mined as follows : 

tr w i 2 

#,, = -* 



*- v a - 

Pn 

Multiplying severally by p l a l) ... p n a n , and adding, we have 

H^r,a = U aa . 

* Trans. Roy. Irish Acad., Vol. xxx. p. 573 (1894). 



286] DEVELOPMENTS OF THE DYNAMICAL THEORY. 305 

Multiplying similarly by p^P^, ...p n /3 n , and adding, 



Eliminating H, we find 

UaaVTpr, - Uafi Srar, = 0. 

We may also prove this formula by physical consideration. Let a, /3 be 
the two screws which correspond, as instantaneous screws, to 77 and , as 
impulsive screws. 

Let us take on the cylindroid a, @, a screw 9, which is conjugate to a 
with respect to inertia ( 81). Then, by known principles, the screw 6 so 
defined must be reciprocal to 77. 



Hence Prf& + ... +p n r) n n = Q. 

As, however, a and 6 are conjugate, we have 



also, since 6 is co-cylindroidal with a and , there must be relations of the 
kind 



Substituting these in the two previous equations, we get 

= ; 

= ; 
whence, as before, 

UaatVpT, U a ptff ayl = 0. 



286. Twist Velocity acquired by an Impulse. 

From the fact that the twist velocity a. acquired by a free body in 
consequence of an impulsive wrench of unit intensity on a screw 77 is 
expressed ( 280) by the equation 

cos (otr?) 
a = - ^- L 

Pa 

we see that the second of the two formulae of 281 may be expressed thus : 



The proof thus given of this expression has assumed that the body is quite 
free. 

It is however a remarkable fact that this formula holds good whatever 
be the constraints to which the body is submitted. If the body receive the 
unit impulsive wrench on a screw 77, the body will commence to twist about 

. 20 



306 THE THEORY OF SCREWS. [286- 

a screw a. But the initial velocity of the body in this case will not generally 
be cos (a.rj) +PO.- It may be easily shown to be 



But we have also 



whence in all cases 

/3-&PT, = O-57 a f . 

This formula is therefore much more general besides being more concise 
than that of 281. 

287. System with Two Degrees of Freedom. 

Let A, B, C, X, &c., and A , B , C , X , &c., be two homographic systems 
of points on a circle. These correspond respectively to two homographic 
systems of screws on the cylindroid according to the method of representation 
in Chap. XII. Then it is known, from geometrical principles, that if any 
two pairs, such as A, A and B, B , be taken, the lines AB , BA intersect 
on a definite straight line, which is the axis of the homography. 

In general this axis may occupy any position whatever ; if, however, it 
should pass through 0, the pole of the axis of pitch, then the homography 
will assume a special type which it is our object to investigate. 

In the first place, we may notice that under these circumstances the 
homography possesses the following characteristic : 

Let A, B be two screws, and A , B their two correspondents ; then, if A 
be reciprocal to B , B must be reciprocal to A . 

For in this case AB must pass through 0, and therefore BA must pass 
through also, i.e. B and A must be reciprocal. 

This cross relation suggests a name for the particular species of homo 
graphy now before us. The form of the letter ^ indicates so naturally the 
kind of relation, that I have found it convenient to refer to this type of 
homography as Chiastic. No doubt, in the present illustration I am only 
discussing the case of two degrees of freedom, but we shall presently see 
that chiastic homography is significant throughout the whole theory. 

288. A Geometrical Proof. 

It is known that in the circular representation the virtual coefficient of 
two screws is proportional to the perpendicular distance of their chord from 
the pole of the axis of pitch ( 61). 



290] DEVELOPMENTS OF THE DYNAMICAL THEORY. 307 

Let a, ft, 7 be three screws of one system, and let 77, f, be the three 
corresponding screws, and, as usual, let r a f represent the virtual coefficient 
of a and . Then whenever the homography is chiastic : 



This is geometrically demonstrated when the following theorem is 
proved : 

If six points be inscribed on a circle, then the continued product of 
the three perpendiculars let fall from any point in a Pascal line formed 
from these six points upon three alternate sides of the corresponding 
hexagon is equal to the continued product of the three perpendiculars let 
fall from the same point on the other three sides. 

Let act , ft ft , 77 be the three pairs of sides, and write the equation 

afty = a ft y , 

then this represents a cubic curve through the nine points a a , aft , ..., and 
this cubic can only be the circle and the Pascal line. 

289. Construction of Chiastic Homography on the Cylindroid. 

It is first obvious that, if two corresponding pairs of screws be arbitrarily 
selected, it will always be possible to devise one chiastic homography of 
which those two pairs are corresponding members. The circular construction 
shows this at once for, join AB and A B, they intersect at T, then the line 
TO is the homographic axis, and the correspondent to X is found by drawing 
A X, and then AX through the intersection of A X and OT. 

290. Homographic Systems on Two Cylindroids. 

The fundamental theorem for the two cylindroids is thus expressed : 

Take any two screws, ct and ft, on one cylindroid, and any two screws, 
77 and , on the other, it will then be possible to inscribe one, and in general 
only one chiastic homography on the two surfaces, such that a and rj shall 
be correspondents, and also ft and . 

For, write the general equation 



If, then, a, ft, 77, are known, and if 7 be chosen arbitrarily on the first 
cylindroid, it will then be always possible to find one, but only one, screw 
on the second cylindroid which satisfies the required condition. 

If a body had two degrees of freedom expressed by a cylindroid A, and 
if an arbitrary cylindroid B were taken, then an impulsive wrench ad 
ministered by any screw on B would make the body commence to twist 

202 



308 THE THEORY OF SCREWS. [290- 

about some corresponding screw on A, and the two systems of screws would 
have chiastic homography. If the body were given both in constitution 
and in position, then, of course, there would be nothing arbitrary in the 
choice of the corresponding screws. Suppose, however, that a screw i) had 
been chosen arbitrarily on B to correspond to a screw a on A, it would 
then be generally possible to design and place a rigid body so that it should 
begin to twist about a in consequence of the impulse on ??. There would, 
however, be no arbitrary element remaining in the homography. Thus, we 
see that, while for homography, in general, three pairs of correspondents 
can be arbitrarily assigned, there can only be two pairs so assigned for 
chiastic homography, while for such a particular type as that which relates 
to impulsive screws and the corresponding instantaneous screws, only one 
pair can be arbitrarily chosen. 

291. Case of Normal Cylindroids. 

We have already had occasion ( 118), to remark on the curious relation 
ship of two cylindroids when a screw can be found on either cylindroid which 
is reciprocal to all the screws on the other. If, for the moment, we speak 
of two such cylindroids as " normal," then we have the following theorem : 

Any homography of the screws on two cylindroids must be chiastic if 
the two cylindroids are normal. 

Let a, /3, 7 be any three screws on one cylindroid, and 77, , any three 
screws on the other ; then, since the cylindroids are normal, we have 



whence we obtain 

GJ >f ( /nr af 5J 0i) 5 J "yf sf at af P 1s yr,) = , 

unless therefore is reciprocal to /3, we must have 

SJ a^^ SJ-yf ^a^ft^yyt = 0. 

If, however, had been reciprocal to /3, then one of these screws (suppose /3) 
must have been the screw on its cylindroid reciprocal to the entire group of 
screws on the other cylindroid. In this case we must have 

5% = ; ts p$ = 0, 
so that even in this case it would still remain true that 

. = 0. 



It is, indeed, a noteworthy circumstance that, for any and every three pairs 
of screws on two normal cylindroids, the relation just written must be 
fulfilled. 



292] DEVELOPMENTS OF THE DYNAMICAL THEORY. 309 

In general, when two pairs of screws are given on two cylindroids, the 
chiastic homography between the surfaces is determined. If, however, it 
were possible to determine two chiastic homographies having two pairs in 
common, then every homography is chiastic, and the cylindroids are normal. 

Let a, 77 and ft, be the two pairs of correspondents, and let 7 have the 
correspondents and , then we have 



whence 

" 

i.e. the two cylindroids are normal. 

292. General Conditions of Chiastic Homography. 

We shall now discuss the relations of chiastic homography between two 
systems of screws in the same w-system. The first point to be demonstrated 
is, that in such a case every pair of the double screws are reciprocal. 

Take a and (3 as two of the double screws, and 77 and will coincide 
with them ; whence the general condition, 



becomes 



= 0. 



One or other of these factors must be zero. We have to show that in general 
it is impossible for 



to vanish. 

For, take 7 reciprocal to a but not to ft, then r av = ; but OT^ Y is not 
zero, and therefore -sr a f would have to be zero ; in other words f must be 
reciprocal to a. But this cannot generally be the case, and hence the other 
factor must vanish, that is 

UTaft = 0. 

In like manner it can be shown that every pair of the double screws must 
be reciprocal. 

Conversely it can be shown that if the double screws of two homo- 
graphic systems are co-reciprocal, then the homography is chiastic. 

Let the w-double screws of the two systems be taken as the screws of 
reference ; then if one screw in one system be denoted by the co-ordinates 

!, ... On, 



310 THE THEORY OF SCREWS. [292, 

its correspondent in the other system will be 

XAj !, ... \h n ct n . 
Similarly, the correspondent to 

&, ... /3 n 
will have for its co-ordinates 

Mi/Si, Mn/3n, 

and the correspondent to 

7i> 7n 
will have for its co-ordinates 



where A,, /n, v, are the constants requisite to make the co-ordinates fulfil the 
fundamental conditions as to dimensions. 

We thus compute 



and similarly for the other terms. 

Whence, by substitution, we find the following equation identically 
satisfied : 



It may be noted that, in a three-system, two homographies are chiastic 
when, in the plane representation by points, the double points of the two 
systems form a triangle which is self-conjugate with respect to the pitch 
conic. 

293. Origin of the formulae of 281*. 

Let a be a screw about which a free rigid body is made to twist in 
consequence of an impulsive wrench administered on some other screw 77. 
Except in the case where a and 77 are reciprocal, it will always be possible 
(in many different ways) to design and place a rigid body so that two 
arbitrarily chosen screws a and 77 will possess the required relation. 

Let now /3 and be two other screws (not reciprocal) : we may consider 
the question as to whether a rigid body can be designed and placed so that 
a shall be the instantaneous screw corresponding to 77 as an impulsive screw, 
while ft bears the same relation to . 

It is easy to see that it will not generally be possible for a, /3, 77, to 
stand in the required relations. For, taking a and /3 as given, there are five 

* Proceedings of the Cambridge Phil. Soc. Vol. ix. Part iii. p. 193. 



293] DEVELOPMENTS OF THE DYNAMICAL THEORY. 311 

disposable quantities in the choice of rj, and five more in the choice of f. 
We ought, therefore, to have ten disposable co-ordinates for the designing 
and the placing of the rigid body. But there are not so many. We have 
three for the co-ordinates of its centre of gravity, three for the direction of 
its principal axes, and three more for the radii of gyration. The other 
circumstances of the rigid body are of no account for our present purpose. 

It thus appears that if the four screws had been chosen arbitrarily we 
should have ten conditions to satisfy, and only nine disposable co-ordinates. 
It is hence plain that the four screws cannot be chosen quite arbitrarily. 
They must be in some way restricted. We can show as follows that these 
restrictions are not fewer than two. 

Draw a cylindroid A through a, fi, and another cylindroid P through rj. 
Then an impulsive wrench about any screw &> on P will make the body 
twist about some screw on A. As eo moves over P, so will its corre 
spondent 6 travel over A. It is shown in 125 that any four screws on 
P will be equianharmonic with their four correspondents on A, and that 
consequently the two systems are homographic. 

In general, to establish the homography of two cylindroids, three cor 
responding pairs of screws must be given ; and, of course, there could be 
a triply infinite variety in the possible homographies. It is, however, a 
somewhat remarkable fact that in the particular homography with which 
we are concerned there is no arbitrary element. The fact that the rigid 
body is supposed quite free distinguishes this special case from the more 
general one of 290. Given the cylindroids A and P, then, without any 
other considerations whatever, all the corresponding pairs are determined. 
This is first to be proved. 

If the mass be one unit, and the intensity of the impulsive wrench on &> 
be one unit, then the twist velocity acquired by 6 is ( 280) 

cos (#&>) 

~~^~ 

where cos(#o>) denotes the cosine of the angle between the two screws 
and ft>, and where p g is the pitch of 6. If, therefore, p g be zero, then cos (#o>) 
must be zero. In other words, the two impulsive screws co ly o> 2 on P, which 
correspond to the two screws of zero pitch d l , 6 2 on A, must be at right 
angles to them, respectively. This will in general identify the correspondents 
on P to two known screws on A. 

We have thus ascertained two pairs of correspondents, and we can now 
determine a third pair. For if o> 3 be a screw on P reciprocal to # 2 , then its 
correspondent 3 will be reciprocal to o> 2 . Thus we have three pairs 6 lt 2 , # 3 
on A, and their three correspondents co l , &> 2 , o>, on P. This establishes the 



312 THE THEORY OF SCREWS. [293- 

homography, and the correspondent 6 to any other screw &> is assigned by 

the condition that the anharmonic ratio of (a^^co is the same as that of 
a a A A 

1/11/2(73(7. 

Reverting to our original screws a and rj, ft and , we now see that they 
must fulfil the conditions 



when the quantities in the brackets denote the anharmonic ratios. 

It can be shown that these equations lead to the formulae of 281. 

294. Exception to be noted. 

We have proved in the last article an instructive theorem which declares 
that when two cylindroids are given it is generally possible in one way, but 
in only one way, to correlate the several pairs of screws on the two surfaces, 
so that when a certain free rigid body received an impulse about the screw 
on one cylindroid, movement would commence by a twisting of the body 
about its correspondent on the other cylindroid. It is, however, easily seen 
that in one particular case the construction for correlation breaks down. 
The exception arises whenever the principal planes of the two cylindroids 
are at right angles. 

The two correspondents on P to the zero-pitch screws on A had been 
chosen from the property that when p a is zero the impulsive wrench must be 
perpendicular to a. We thus take the two screws on P which are respec 
tively perpendicular to the two zero-pitch screws. But suppose there are 
not two screws on P which are perpendicular to the two zero-pitch screws on 
A. Suppose in fact that there is one screw on P which is parallel to the 
nodal axis of A, then the construction fails. We would thus have a single 
screw on P with two corresponding instantaneous screws for the same body. 
This is of course impossible, and accordingly in this particular case, which 
happens when the principal planes of P and A are rectangular, it is impos 
sible to adjust the correspondence. 

295. Impulsive and Instantaneous Cylindroids. 

Let X, X be two screws on a cylindroid whereof a and jB are the two 
principal screws. 

Let 0, 6 be the angles which X and X respectively make with a. 

We shall take the six absolute screws of inertia as the screws of reference 
and we have as the co-ordinates of X 

! cos 6 + & sin 6, . . . a cos 6 + /3 6 sin 6, 



295] DEVELOPMENTS OF THE DYNAMICAL THEORY. 313 

and of X 

! cos + & sin , ... a 6 cos & + /3 6 sin &. 

In like manner, let p and p be two screws on a cylindroid, of which the 
two principal screws are rj and f. 

Let <f>, < be the angles which p and p make respectively with . 
Then the co-ordinates of p are 

r) l cos </> + f j sin (j), ... rj 6 cos </> + 6 sin <, 
and of p 

7?! cos </> + j sin </> , ... 77,3 cos + 6 sin < , &c. 

We shall now suppose that the two cylindroids a, /3 and 77, are so 
circumstanced that the latter is the locus of the impulsive wrenches cor 
responding to the several instantaneous screws on the former with respect 
to the rigid body which is to be regarded as absolutely free. We shall 
further assume that p is the impulsive screw which has X as its instantaneous 
screw, and that the relation of p to X is of the same nature. 

If, however, the four screws X, X , p, p possess the relations thus indi 
cated, it is necessary that they satisfy the conditions already proved ( 281). 
These are two-fold, and they are expressed by the following equations, as 
already shown : 

cos (\p ) + P,*,. cos (\p f ) = 2r A v, 



cos (\p) cos 

p\ p\ 



We shall arbitrarily choose X and p, so as to satisfy the conditions 

A /P = 0, tnv = 0, 



and thus the second of the two equations is satisfied. These two equations 
will give & as a function of <, and </> as a function of 6. We can thus 
eliminate & and < from the first of the two equations, and the result will 
be a relation connecting 6 and $. This equation will exhibit the relation 
between any instantaneous screw 6 on one cylindroid, and the corresponding 
impulsive screw </> on the other. 

It will be observed that when the two cylindroids are given, the required 
equation is completely denned. The homographic relations of p and X is 
thus completely determined by the geometrical relations of the two cylin 
droids. 



314 



THE THEORY OF SCREWS. 



[295, 



The calculation* presents no difficulty and the result is as follows : 



= cos 6 cos (f> 



+ cos 6 sin (f> 



+ sin 6 cos </> 



+ sin 9 sin 



+p a cos 



cos 



- -S7 a cos 



cos (a??) [r a , cos (a) *3- j cos (a?;)] 
***, [cos (a) cos (77) - cos (a?;) cos 



+p a cos 



r a , cos 



- OT a cos 



cos (a) [r a , cos (a) nr a f cos (CM?)] 
-GTpt [cos (a) cos (/ify) cos (arj) cos 
cos (at)) \ixfr cos (a) -BT^ cos (a?;)] 



+ p a COS (77) [tjp, cos 

a, [cos (a) cos (/ify) cos (atj) cos 



+ PP cos (a^) [or^, cos (a^) tsfc cos (a??)] 



cos / r p , cos - ^ j cos 
_+ ^-B3-a| [cos (af ) cos (firf) cos (a??) cos 



296. An exceptional Case. 

A few remarks should be made on the failure of the correspondence 
when the principal planes of the two cylindroids are at right angles ( 294). 
It will be noted that though this equation suffers a slight reduction when 
the principal planes of the two cylindroids are at right angles yet it does 
not become evanescent or impossible. For any value of 6 defining a screw 
on one cylindroid, the equation provides a value of < for the correspondent 
on the other cylindroid. Thus we seem to meet with a contradiction, for 
while the argument of 294 shows that in such a case the homography 
is impossible, yet the homographic equation seemed to show that it was 
possible and indeed fixed the pairs of correspondents with absolute 
definiteness. 

It is certainly true that if two cylindroids A and P admit of the cor 
relation of their screws into pairs whereof those on P are impulsive screws 
and those on A are instantaneous screws, the pairs of screws by which the 
homographic equation is satisfied will stand to each other in the desired 
relation. If, however, the screws on two cylindroids be correlated into 
pairs in accordance with the indications of the homographic equation, 
though it will generally be true that there may be corresponding impulsive 
screws and instantaneous screws, yet in the case where the principal planes 
of the cylindroids are at right angles no such inference can be drawn. 

The case is a somewhat curious one. It will be seen that the calculation 



See Trans. Eoy. Irish Acad. Vol. xxx. p. 112 (1894). 



296] DEVELOPMENTS OF THE DYNAMICAL THEORY. 315 

of the homographic equation is based on the fact that if X, \ be two 
instantaneous screws and p, p the corresponding impulsive screws, then 
the formula 

COS (X + P * t , x COS (\p f ) = 2tsr AX 



, , t , x 

cos (X/o) cos \Kff) 

must be satisfied. 

And generally it is satisfied. In the case of two cylindroids with normal 
planes it is however easy to show that there are certain pairs of screws for 
which this formula cannot obtain. 

For in such a case there is one screw X on A which is perpendicular to 
every screw on P, so that whatever be the p corresponding to X, 

cos (\p) 0. 

Since no other screw X can be perpendicular to any screw on P we cannot 
have either 

cos (X p), or cos (A///), zero. 

Hence this equation cannot be satisfied and the argument that the homo- 
graphic equation defines corresponding pairs is in this case invalid. 

We might have explained the matter in the following manner. 

When the principal planes of A and P are normal there is one screw X 
on A which is perpendicular to all the screws on P. If therefore the two 
cylindroids were to be impulsive and instantaneous, there must be a screw 
on P which corresponds to X. It can be shown in general ( 301) that 

d\ = p\ tan (X0) 

when dx is the perpendicular from the centre of gravity on X; it follows 
that when (A.0) = 90 we must have either p^ zero or d\ infinite. 

But of course it will not generally be the case that X. happens to be one 
of the screws of zero pitch on A. Hence we are reduced to the other 
alternative 

d*. = infinity. 

This means that the centre of gravity is to be at infinity. 

But when the centre of gravity of the body is at infinity a remarkable 
consequence follows. All the instantaneous screws must be parallel. 

For if 6 be the impulsive wrench corresponding to X as the instantaneous 
screw, then we know that 

di=p*. tan(X0), 
and that the centre of gravity lies in a right line parallel to X and distant 






316 THE THEORY OF SCREWS. [296- 

from it by d^. In like manner if </> be an impulsive screw corresponding to 
yu, as instantaneous screw we have another locus parallel to p for the centre 
of gravity. 

But as the centre of gravity is at infinity these two loci must there 
intersect, i.e. they must be parallel and so must X and fi, and hence all 
instantaneous screws must be parallel. 

Thus we see that all the screws on A must be parallel, i.e. that A must 
have degraded into an extreme type of cylindroid. 

297. Another extreme Case. 

.Given any two cylindroids A and P it is, as we have seen, generally 
possible to correlate in one way the several screws on A to those on P so 
that an impulsive wrench given to a certain rigid body about any screw on 
P would make that body commence to move by twisting about its cor 
respondent on A. One case of failure has just been discussed. The case 
now to be considered is not indeed one of failure but one in which any 
two pairs of screws on A and P will stand to each other in the desired 
relations. 

Suppose that A and P happened to fulfil the single condition that each 
of them shall contain one screw which is reciprocal to the other cylindroid. 
We have called the cylindroids so circumstanced " normal." 

Let A, be the screw on A which is reciprocal to every screw on P. If 
then P and A are to stand to each other in the required relation, A. must 
be reciprocal to its impulsive screw. But this is only possible on one 
condition. The mass of the body must be zero. In that case, if there is 
no mass involved any one of the screws on P may be the impulsive screw 
corresponding to any one of the screws on A . 

Here again the question arises as to what becomes of the homographic 
equation which defines so precisely the screw on P which corresponds to 
the screws on A ( 295). It might have been expected that in the case 
of two normal cylindroids this homographic equation should become evan 
escent. But it does not do so. 

But there is no real contradiction. The greater includes the less. 
If every screw on P will suit as correspondent every screw on A then d 
fortiori will the pairs indicated by the homography fulfil the conditions 
requisite. 

That any two pairs of screws will be correspondents in this case is 
obvious from the following. 



298] DEVELOPMENTS OF THE DYNAMICAL THEORY. 317 

Let X be the screw on A which is reciprocal to P, 
S .................. P ........................ A. 

Then any screw //, on A and any screw cj> on P fulfil the conditions 

WA$ = 0, TOV,, = 0. 

Hence < is the impulsive screw corresponding to fj, as the instantaneous 
screw. 

298. Three Pairs of Correspondents. 

Let a, T\\ ft, J; ; 7, be three pairs of impulsive and instantaneous screws ; 
let 9, </> be another pair. Then, if we denote by L a p = 0, and M a p = 0, the 
two fundamental equations 



C08 



Pa Pfi 

cos (XT ** ~ cos 



we shall obtain six equations of the type 

L 6a = 0, L 9li = 0, L^ = 0, 
Jlfe a = 0, Jlftf = 0, M ey = 0. 

From these six it might be thought that <f> lt ... < 6 could be eliminated, 
and thus it would, at first sight, seem that there must be an equation for 6 
to satisfy. It is, however, obvious that there can be no such condition, for 6 
can of course be chosen arbitrarily. The fact is, that these equations have 
a peculiar character which precludes the ordinary algebraical inference. 

Since ,;; ft, ; 7, ; are three pairs of screws, fulfilling the necessary 
six conditions, a rigid body can be adjusted to them so that they are 
respectively impulsive and instantaneous. We take the six principal screws 
of inertia of this body as the screws of reference. We thus have, where 
p a , pp, p y are certain factors, 



By putting the co-ordinates in this form, we imply that they satisfy the 
six equations of condition above written. 

Substituting the co-ordinates in L 0a = 0, we get 
= + (a, + a 2 ) (pefa + p 9 fa) + (a, + a,) (p e <f> 3 + p e $ t ) + ( + a.) (p e <j> t 
+ (0i + 0a) (ai - 2> + (0 + 04> (b<*> ~ 6*4) + (0 5 + 6 ) (c 8 - ca 6 ) 
- 2 (aa^ - a^6 2 ) - 2 (6o0 - ba,e 4 ) - 2 (c 5 ^ 5 - ca,0). 



318 



THE THEORY OF SCREWS. 



[298, 



Let 



! a6 l = X ly 



4 4- 6# 4 = X t , 
c0 6 = X 6 , 



and the equation becomes 

= (! + 2 ) (X, + X,} + ( 3 + 4 ) (X 3 
and the two other L equations give 

= (A + &) (X, + X 2 ) + (0, + &) (X 3 

= (71 + 7 



5 + 6 ) (X 5 + X 6 ) ; 



/3 6 ) (X, 



a) + (73 + 74) (-^s + -^4) + (75 4- 7 6 ) (X s + X 6 ). 

If we eliminate X^ + X i} X 3 + X t , X S + X S from these equations, we 
should have 

= Otj + 85 0(3 + 4 Qfg + Ct 6 

A 4/3, & 4- A & 4 
7i +7s 7s + 74 7s + 7e 

But this would only be the case if a, /3, 7 were parallel to a plane, which is 
not generally true. Therefore, we can only satisfy these equations, under 
ordinary circumstances, by the assumption 



In like manner, the equations of the M type give 

Pa^Or, = 0, 



Substituting, in the first of these, we get 



- p a (017! 0j - a?? 2 ^ 2 4- br} 3 3 - 6774 4 + cv) s s - cy^e) = ; 
which reduces to 

ac^Zj - a 2 Z 2 + ba 3 X s - 6a 4 Z 4 + ca s X s - ca 6 X e = ; 

but we have already seen that X^ -+ X 2 = 0, &c., whence we obtain 
Xi (a! + aoz) + X 3 (6a 3 + 6 4 ) + X 5 (c 6 + ca 6 ) = ; 

with the similar equations 

X l (a/9, + a/8 2 ) + X 3 (b/3 3 + bfr) + X 5 (cyS 5 + c&) = 0, 
^! (ay l + ay 2 ) + X 3 (by, + by 4 ) + X 5 (c% + cy 6 ) = 0. 

These prove that, unless a, @, y be parallel to a plane, we must have 



299] DEVELOPMENTS OF THE DYNAMICAL THEORY. 319 

X 1 = 0, X 3 = 0, X s = 0. Combining these conditions with the last, we draw 
the general conclusion that 



or 



&c. 



Thus we demonstrate that if a pair of screws 6, $> satisfy the six conditions, 
they stand to each other in the relation of impulsive screws and instantaneous 
screws. 

299. Cylindroid Reduced to a Plane. 

Suppose that the family of rigid bodies be found which make a, 77 and 
/?, impulsive and instantaneous. Let there be any third screw, 7, and let 
us seek for the locus of its impulsive screw, , for all the different rigid 
bodies of the family. 

must satisfy the four equations 

cos (777) + / cos (a?) = 2sr ay , 



cos (arj) cos 



-^008(^0 = 
COS (yC) 



cos (ar;) Y>) cos 



As there are four linear equations in the coordinates of , we have the 
following theorem. 

If a, 77 and /3, f be given pairs of impulsive and instantaneous screws, 
then the locus of the impulsive screw corresponding to 7, as an instan 
taneous screw, is a cylindroid. 

But this cylindroid is of a special type. It is indeed a plane surface 
rather than a cubic. The equations for can have this form : 

cos (a?) = A cos (yf), vr a( = C cos (y ), 

COS (f ) = B COS (y), CT^ = D COS (yf ), 

in which A, B, C, D are constants. 



The fact that cos (af) and cos (7^) have one fixed ratio, and cos (/3) and 
cos (yf) another, shows that the direction of is fixed. The cylindroidal 
locus of , therefore, degenerates to a system of parallel lines. 

At first it may seem surprising to find that CT of is constant. But the 



320 THE THEORY OF SCREWS. [299- 

necessity for this arrangement is thus shown. If not constant, then there 
would generally have been some screw , for which tsr a ^ was zero. In this 
case, of course, ^ yn would be generally zero also. But 7 and tj being both 
given, this is of course not generally true. The only escape is for r af 
to be constant. 



300. A difficulty removed. 

Given a and 77, @ and and also 7, then the plane of is determined 
from the equations of the last article. 

As OT a and ts^ are constant, both a and /3 must be parallel to the plane 
already considered. But as an impulsive screw could not be reciprocal to an 
instantaneous screw, it would seem that w y ^ must never be zero, but this 
condition can only be fulfilled by requiring that must be parallel to 
the same plane. Whence a, /3, 7 must be parallel to the same plane. But 
these three screws are quite arbitrary. Here then would seem to be a 
contradiction. 

The difficulty can be explained as follows : 

Each rigid body, which conformed to the condition that a, /3 and 77, f 
shall be two pairs of corresponding impulsive and instantaneous screws, will 
have a different screw corresponding to a given screw 7. Thus, among the 
various screws , in the degraded cylindroid, each will correspond to one 
rigid body. In general, of course, it would be impossible for f to be 
reciprocal to 7. It would be impossible for an impulsive wrench to make a 
body twist about a screw reciprocal thereto. Nevertheless, it seemed certain 
that, in general, there must be a screw reciprocal to 7. For otherwise, 
a, /3, 7 should be all parallel to a plane, which, of course, is not generally 
true. If, however, a, or b, or o were zero, then the body will have no 
mass ; consequently no impulse would be necessary to set it in motion. 
This clearly is the case when is reciprocal to 7. We have thus got over 
the difficulty. and 7 are reciprocal, in the case when the rigid body is 
such that a, or 6, or c is zero. 

301. Two Geometrical Theorems. 

The perpendicular from the centre of gravity on any instantaneous screw 
is parallel to the shortest distance between that instantaneous screw and the 
corresponding impulsive screw. 

The perpendicular from the centre of gravity on any instantaneous screw 
is equal to the product of the pitch of that screw, and the tangent of the angle 
between it and the corresponding impulsive screw. 



301] DEVELOPMENTS OF THE DYNAMICAL THEORY. 321 

Let a be the instantaneous screw and d a the length of the perpendicular 
thereon from the centre of gravity. If cos A,, cos /i, cos v be the direction 
cosines of d a then 

d a cos X = ( 5 - a 6 ) (a, + 4 ) c - ( 5 + a 6 ) ( 3 - o 4 ) b, 
d a cos /*=(!- 2 ) (a, + a e )a- fa + o^) (a 5 - 6 ) c, 
d a cos v = ( 3 4 ) (ttj + a 2 ) & ( 3 + 4 ) (i 2 ) a. 

But if 77 is the impulsive screw corresponding to a as the instantaneous 
screw we have 

! = - ^-\*hi -C& 2 = - ^ N^"> &C., &C., 
cos (a?;) cos (a?;) 

whence 

*) 

da COS X = CQS Y . ((T/., + 77 fi ) (173 + 4 ) - ( 5 + e) (173 + 1; 4 )), 
2) 

rf a cos /^ = / ?. ((% + 7? 2 ) ( 5 + 6 ) - (! + 2 ) (775 + 77 6 )), 



d COS " = ^T~\ (^s + ^4> (i + On) - (s + 4 ) (77! + 77 2 )). 
COo \Jjiij} 

But 

(% + %) ( 3 + 4 ) - ( B + 6> (?3 + 7 4 ) = sin (077) cos X , 

with similar expressions for sin (0(77) cos /* and sin (377) cos v where cos X , 
cos p, and cos v are the direction cosines of the common perpendicular to 
and 77. We have therefore 

v 

d a cos X = v r sin (an) cos X , 
cos 



in 

d a coKu,= ~ . sin (77)cos /A 
cos (at}) 

rn 

d a cos v = *? - sin (a77) cos v, 
cos (277) 

whence 

cos X = cos X ; cos /it = cos pf ; cos v cos v ; 

and d a = p a tan (77), 

which proves the theorems. 



B. 21 



CHAPTER XXII. 

THE GEOMETRICAL THEORY*. 

302. Preliminary. 

IT will be remembered how Poinsot advanced our knowledge of the 
dynamics of a rigid system by a beautiful geometrical theory of the rotation 
of a rigid body about a fixed point. We now specially refer to the geometrical 
construction by which he determined the instantaneous axis about which 
the body commenced to rotate when the plane of the instantaneous couple 
was given. 

We may enunciate with a generality, increasing in successive steps, the 
problem which, in its simplest form, Poinsot had made classical. From the 
case of a rigid body which is constrained to rotate about a fixed point we 
advance to the wider conception of a body which has three degrees of 
freedom of the most general type. We can generalize this again into the 
case in which the body, instead of having a definite number of degrees of 
freedom has any number of such degrees. The range extends from the 
first or lowest degree, where the only movement of which the body is 
capable is that of twisting about a single fixed screw, up to the case in 
which the body being perfectly free, or in other words, having six degrees of 
freedom, is able to twist about every screw in space. It will, of course, be 
borne in mind that only small movements are to be understood. 

In a corresponding manner we may generalize the forces applied to the 
body. In the problem solved by Poinsot the effective forces are equivalent 
to a couple solely. For the reaction of the fixed point is capable of reducing 
any system of forces whatever to a couple. But in the more generalized 
problems with which the theory of screws is concerned, we do not restrict 
the forces to the specialized pair which form a couple. We shall assume 
that the forces are of the most general type and represented by a wrench 
upon a screw. Thus, by generalizing the freedom of the rigid body, as well 
as the forces which act upon it, we may investigate the geometrical theory of 
the motion when a rigid body of the most general type, possessing a certain 
number of degrees of freedom of the most general type, is disturbed from a 

* Trans. Royal Irish Acad. Vol. xxi. (1897) p. 185. 



302, 303] THE GEOMETRICAL THEORY. 323 

position of rest by an impulsive system of forces of the most general type. 
This is the object of the present chapter. 

303. One Fair of Impulsive and Instantaneous Screws. 

Let it be supposed that nothing is known of the position, mass, or other 
circumstances of an unconstrained rigid body save what can be deduced 
from the fact that, when struck from a position of rest by an impulsive 
wrench on a specified screw 77, the effect is to make the body commence to 
move by twisting around a specified screw a. 

As a, like every other screw, is defined by five coordinates, the knowledge 
of this screw gives us five data towards the nine data that are required for 
the complete specification of the rigid body and its position. 

We have first to prove that the five elements which can be thence 
inferred with respect to the rigid body are in general 

(1) A diameter of the momental ellipsoid. 

This is clearly equivalent to two elements, inasmuch as it restricts 
the position of the centre of gravity to a determinate straight line. 

(2) The radius of gyration about this diameter. 

This is, of course, one element. 

(3) A straight line in the plane conjugate to that diameter. 

A point in the plane would have been one element, but a straight 
line in the plane is equivalent to two. If the centre of gravity were 
also known, we should at once be able to draw the conjugate plane. 

Draw a plane through both the instantaneous screw a and the common 
perpendicular to a and 77. Then the centre of gravity of the rigid body 
must lie in that plane ( 301). It was also shown that if p a be the pitch of 
a, and if (ar)) represent the angle between a and 77, then the perpendicular 
distance of the centre of gravity from a. will be expressed by p a tan ((277) 
( 301). This expression is completely known since a and 77 are known. 
Thus we find that the centre of gravity must lie in a determinate ray 
parallel to a. There will be no ambiguity as to the side on which this 
straight line lies if it be observed that it must pass between a and the point 
in which 77 is met by the common perpendicular to 77 and a. In this manner 
from knowing a and 77 we discover a diameter of the momental ellipsoid. 

If a be the twist velocity with which a rigid body of mass M is twisting 
about any screw a. If 77 be the corresponding impulsive screw, and if Ts ar> 
denote as usual the virtual coefficient of a and 77, then it is proved in 279 
that the kinetic energy of the body 

MCt 2 -, r GJ an- 

cos (077) 

212 



324 THE THEORY OF SCREWS. [303- 

We can now determine the value of p a * where p a is the radius of gyration 
about an axis parallel to a through the centre of gravity. For the kinetic 
energy is obviously 

P/d a (/t>a a +^. 2 + a 2 )- 

By equating the two expressions we have 

a 2 = 2 



cos 

But when a and rj are known the three terms on the right-hand side of 
this equation are determined. Thus we learn the radius of gyration on the 
diameter parallel to a. 

It remains to show how a certain straight line in the plane which is 
conjugate to this diameter in the momental ellipsoid is also determined. 
Let a screw 6, of zero pitch, be placed on that known diameter of the 
momental ellipsoid which is parallel to or. Draw a cylindroid through the 
two screws 6 and 77. Let </> be the other screw of the zero pitch, which will 
in general be found on the same cylindroid. 

We could replace the original impulsive wrench on 77 by its two com 
ponent wrenches on any two screws of the cylindroid. We choose for this 
purpose the two screws of zero pitch 6 and <. Thus we replace the wrench 
on 77 by two forces, whose joint effect is identical with the effect that would 
have been produced by the wrench on 77. 

As to the force along the line 6 it is, from the nature of the con 
struction, directed through the centre of gravity. Such an impulsive force 
would produce a velocity of translation, but it could have no effect in pro 
ducing a rotation. The rotatory part of the initial twist velocity must there 
fore be solely the result of the impulsive force on <. 

But when an impulsive force is applied to a quiescent rigid body we 
know, from Poinsot s theorem, that the rotatory part of the instantaneous 
movement must be about an axis parallel to the direction which is conjugate 
in the momental ellipsoid to the plane which contains both the centre of 
gravity and the impulsive force. It follows that the ray (f> must be situated 
in that plane which is conjugate in the momental ellipsoid to the diameter 
parallel to a. But, as we have already seen, the position of </> is completely 
defined on the known cylindroid on which it lies. We have thus obtained a 
fixed ray in the conjugate plane to a known diameter of the momental 
ellipsoid. 

The three statements at the beginning of this article have therefore been 
established. We have, accordingly, ascertained five distinct geometrical data 
towards the nine which are necessary for the complete specification of the 
rigid body. These five data are inferred solely from our knowledge of a 
single pair of corresponding impulsive and instantaneous screws. 



305] THE GEOMETRICAL THEORY. 325 

304. An Important Exception. 

If p a = 0, then (a?;) is 90, and consequently p a tan (a?;) is indefinite. 
If, therefore, the pitch of the instantaneous screw be zero, then we are no 
longer entitled to locate the centre of gravity in a certain ray. All we know 
is that it lies in the plane through a perpendicular to 77. In general the 
knowledge of the impulsive screw corresponding to a given instantaneous 
screw implies five data, yet this ceases to be the case if p a is zero, for as 77 
must then be perpendicular to a there are really only four independent data 
given when 77 is given. We have, therefore, in this case one element the less 
towards the determination of the rigid body. 

305. Two Pairs of Impulsive and Instantaneous Screws. 

Let us next suppose that we are given a second pair of corresponding 
impulsive and instantaneous screws. We shall examine how much further 
we are enabled to proceed by the help of this additional information towards 
the complete determination of the rigid body in its abstract form. Any data 
in excess of nine, if not actually impossible, would be superfluous. If, 
therefore, we are given a second pair of impulsive and instantaneous screws, 
the five data which they bring cannot be wholly independent of the five data 
brought by the preceding pair. It is therefore plain that the quartet of 
screws forming two pairs of corresponding impulsive screws and instantaneous 
screws cannot be chosen arbitrarily. They must submit to at least one 
purely geometrical condition, so that the number of data independent of each 
other shall not exceed nine. 

It is, however, not so obvious, though it is certainly true, as we found in 
281, that the two pairs of screws must conform not merely to one, but to 
no less than two geometrical conditions. In fact, if 77, be two impulsive 
screws, and if a, /3 be the two corresponding instantaneous screws, then, 
when the body acted upon is perfectly free, the following two formulae must 
be satisfied : 

P Pf* 



cos + cos 



cos (a??) cos 

We can enunciate two geometrical properties of the two pairs of screws, 
which are equivalent to the conditions expressed by these equations. 

In the first place, each of the pairs of screws determines a diameter of 
the momental ellipsoid. The fact that the two diameters, so found, must 
intersect each other, is obviously one geometrical condition imposed on the 
system a, 77 and /3, . 



326 THE THEORY OF SCREWS. [305, 

Let G be this intersection, and draw OP parallel to a and equal to the 
radius of gyration about GP, which we have shown to be known from the 
fact that a and ij are known. Let X be the plane conjugate to GP in the 
moraental ellipsoid, then this plane is also known. 

In like manner, draw GQ parallel to ft and equal to the radius of 
gyration about GQ. Let Y be the plane, conjugate to GQ, in the momental 
ellipsoid. 

Let PI and P 2 be the perpendiculars from P, upon X and Y respec 
tively. 

Let Q l and Q 2 be the perpendiculars from Q, upon X and Y respec 
tively. 

Then, from the properties of the ellipsoid, it is easily shown that 

PI ^2 = Qi Qv 

This is the second geometrical relation between the two pairs of screws 
a, ?; and ft, . Subject to these two geometrical conditions or to the two 
formulse to which they are equivalent the two pairs of screws might be chosen 
arbitrarily. 

As these two relations exist, it is evident that the knowledge of a second 
pair of corresponding impulsive screws and instantaneous screws cannot 
bring five independent data as did the first pair. The second pair can bring 
no more than three. From our knowledge of the two pairs of screws together 
we thus obtain no more than eight data. We are consequently short by 
one of the number requisite for the complete specification of the rigid body 
in its abstract form. 

It follows that there must be a singly infinite number of rigid bodies, 
every one of which will fulfil the necessary conditions with reference to the 
two pairs of screws. For every one of those bodies a is the instantaneous 
screw about which twisting motion would be produced by an impulsive 
wrench on 77. For every one of those bodies ft is the instantaneous screw 
about which twisting motion would be produced by an impulsive wrench 
on f. 

306. A System of Rigid Bodies. 

We have now to study the geometrical relations of the particular system 
of rigid bodies in singly infinite variety which stand to the four screws in the 
relation just specified. 

Draw the cylindroid (a, ft) which passes through the two screws a and ft. 
Draw also the cylindroid (q, ) which passes through the two corresponding 
impulsive screws 17 and It is easily seen that every screw on the first of 
these cylindroids if regarded as an instantaneous screw, with respect to the 
same rigid body, will have its corresponding impulsive screw on the second 






306] THE GEOMETRICAL THEORY. 327 

cylindroid. For any impulsive wrench on (77, ) can be decomposed into 
impulsive wrenches on 77 and . The first of these will generate a twist 
velocity about a. The second will generate a twist velocity about /3. These 
two can only compound into a twist velocity about some other screw on the 
cylindroid (a, /3). This must, therefore, be the instantaneous screw corre 
sponding to the original impulsive wrench on (77, ). 

It is a remarkable point about this part of our subject that, as proved 
in 293, we can now, without any further attention to the rigid body, corre 
late definitely each of the screws on the instantaneous cylindroid with its 
correspondent on the impulsive cylindroid. 

We thus see how, from our knowledge of two pairs of correspondents, we 
can construct the impulsive screw on the cylindroid (77, ) corresponding to 
every screw on the cylindroid (a, /3). 

It has been already explained in the last article how a single known 
pair of corresponding impulsive and instantaneous screws suffice to point 
out a diameter of the momental ellipsoid, and also give its radius of 
gyration. A second pair of screws will give another diameter of the 
momental ellipsoid, and these two diameters give, by their intersection, the 
centre of gravity. As we have an infinite number of corresponding pairs, 
we thus get an infinite number of diameters, all, however, being parallel to 
the principal plane of the instantaneous cylindroid. The radius of gyration 
on each of these diameters is known. Thus we get a section 8 of the 
momental ellipsoid, and we draw any pair of conjugate diameters in that 
section. These diameters, as well as the radius of gyration on each of them, 
are thus definitely fixed. 

When we had only a single pair of corresponding impulsive and instan 
taneous screws, we could still determine one ray in the conjugate plane to 
the diameter parallel to the instantaneous screw. Now that we have further 
ascertained the centre of gravity, the conjugate plane to the diameter, 
parallel to the instantaneous axis, is completely determined. Every pair of 
corresponding impulsive and instantaneous screws will give a conjugate 
plane to the diameter parallel to the instantaneous screw. Thus we know 
the conjugate planes to all the diameters in the plane S. All these planes 
must intersect, in a common ray Q, which is, of course, the conjugate 
direction to the plane S. 

This ray Q might have been otherwise determined. Take one of the two 
screws, of zero pitch, in the impulsive cylindroid (77, ). Then the plane, 
through this screw and the centre of gravity, must, by Poinsot s theorem 
already referred to, be the conjugate plane to some straight line in S. 
Similarly, the plane through the centre of gravity and the other screw of 
zero pitch, on the cylindroid (77, ), will also be the conjugate plane to some 



328 THE THEORY OF SCREWS. [306 

ray in S. Hence, we see that the ray Q must lie in each of the planes so 
constructed, and is therefore determined. In fact, it is merely the transversal 
drawn from the centre of gravity to intersect both the screws of zero pitch 
on the cylindroid (77, ). 

We have thus proved that when two pairs of corresponding impulsive 
screws and instantaneous screws are given, we know the centre of the 
momental ellipsoid, we know the directions of three of its conjugate 
diameters, and we know the radii of gyration on two of those diameters. 
The radius of gyration on the third diameter remains arbitrary. Be that 
radius what it may, the rigid body will still fulfil the condition rendering 
a, ?? ond /3, | respective pairs of instantaneous screws and impulsive screws. 
We had from the first foreseen that the data would only provide eight 
coordinates, while the specification of the body required nine. We now 
learn the nature of the undetermined coordinate. 

It appears from this investigation that, if two pairs of impulsive screws 
and the corresponding instantaneous screws are known, but that if there be 
no other information, the rigid body is indeterminate. It follows that, if an 
impulsive screw be given, the corresponding instantaneous screw will not 
generally be determined. Each of the possible rigid bodies will have a 
different instantaneous screw, though the impulsive screw may be the same. 
It was, however, shown ( 299), that all the instantaneous screws which 
pertain to a given impulsive screw lie on the same cylindroid. It is 
a cylindroid of extreme type, possessing a screw of infinite pitch, and 
degenerating to a plane. 

Even while the body is thus indeterminate, there are, nevertheless, 
a system of impulsive screws which have the same instantaneous screw for 
every rigid body which complies with the expressed conditions. Among 
these are, of course, the several screws on the impulsive cylindroid (rj. f) 
which have each the same corresponding screw on the instantaneous cylin 
droid (a, /3), whatever may be the body of the system to which the impulsive 
wrench is applied. But the pairs of screws on these two cylindroids are 
indeed no more than an infinitesimal part of the total number of pairs of 
screws that are circumstanced in this particular way. We have to show 
that there is a system of screws of the fifth order, such that an impulsive 
wrench on any one of those screws rj will make any body of the system com 
mence to twist about the same screw a. 

As already explained, the system of rigid bodies have a common centre 
of gravity. Any force, directed through the centre of gravity, will produce 
a linear velocity parallel to that force. This will, of course, apply to every 
body of the system. All possible forces, which could be applied to one 
point, form a system of the third order of a very specialized type. Each one 



306] THE GEOMETRICAL THEORY. 329 

of the screws of this system will have, as its instantaneous screw, a screw of 
infinite pitch parallel thereto. We have thus a system of impulsive screws 
of the third order, and a corresponding system of instantaneous screws of 
the third order, the relation between each pair being quite independent of 
whatever particular rigid body of the group the impulsive wrench be 
applied to. 

This system of the third order taken in conjunction with the cylindroid 
(?;, ) will enable us to determine the total system of impulsive screws which 
possess the property in question. Take any screw 6, of zero pitch, passing 
through the centre of gravity, and any screw, (f>, on the cylindroid (w, ). 
We know, of course, as already explained, the instantaneous screws corre 
sponding to 9 and <. Let us call them \, /j,, respectively. Draw the 
cylindroid (6, <), and the cylindroid (X, //,). The latter of these will be the 
locus of the instantaneous screws, corresponding to the screws on the former 
as impulsive screws. From the remarkable property of the two cylindroids, 
so related, it follows that every impulsive screw on (9, <) will have its 
corresponding instantaneous screw on (X, p) definitely fixed. This will be so, 
notwithstanding the arbitrary element remaining in the rigid body. From 
the way in which the cylindroid (9, (f>) was constructed, it is plain that the 
screws belonging to it are members of the system of the fifth order, formed 
by combinations of screws on the cylindroid (rj, ) with screws of the special 
system of the third order passing through the centre of gravity. But all 
the screws of a five-system are reciprocal to a single screw. The five-system 
we are at present considering consists of the screws which are reciprocal to 
that single screw, of zero pitch, which passes through the centre of gravity 
and intersects both the screws, of zero pitch, on the impulsive cylindroid 
(77, ). The corresponding instantaneous screws will also form a system of the 
fifth order, but it will be a system of a specialized type. It will be the result 
of compounding all possible displacements by translation, with all possible 
twists about screws on the cylindroid (a, /3). The resulting system of the 
fifth order consists of all screws, of whatsoever pitch, which fulfil the single 
condition of being perpendicular to the axis of the cylindroid (a, /3). Hence 
we obtain the following theorem : 

If an impulsive cylindroid, and the corresponding instantaneous cylin 
droid, be known, we can construct, from these two cylindroids, and without any 
further information as to the rigid body, two systems of screws of the fifth 
order, such that an impulsive wrench on a given screw of one system will 
produce an instantaneous twist velocity about a determined screw on the other 
system. 

It is interesting to note in what way our knowledge of but two corre 
sponding pairs of impulsive screws and instantaneous screws just fails to 
give complete information with respect to every other pair. If we take any 



330 THE THEORY OF SCREWS. [306, 

ray in space, and assign to it an arbitrary pitch, the screw so formed may be 
regarded as an impulsive screw, and the corresponding instantaneous screw 
will not, in general, be defined. There is, however, a particular pitch for 
each such screw, which will constitute it a member of the system of the 
fifth order. It follows that any ray in space, when it receives the proper 
pitch, will be such that an impulsive wrench thereon would set any one 
of the singly infinite system of rigid bodies twisting about the same 
screw a. 

307. The Geometrical Theory of Three Pairs of Screws. 

We can now show how, when three pairs of corresponding impulsive 
screws and instantaneous screws are given, the instantaneous screw, corre 
sponding to any impulsive screw, is geometrically constructed. 

The solution depends upon the following proposition, which I have set 
down in its general form, though the application to be made of it is somewhat 
specialized. 

Given any two independent systems of screws of the third order, Pand Q. 
Let &) be any screw which does not belong either to P or to Q, then it is 
possible to find in one way, but only in one, a screw 6, belonging to P, and a 
screw <j>, belonging to Q, such that o>, 6 and (f> shall all lie on the same cylin- 
droid. This is proved as follows. 

Draw the system of screws of the third order, P , which is reciprocal to P, 
and the system Q , which is reciprocal to Q. The screws belonging to P , 
and which are at the same time reciprocal to a>, constitute a group reciprocal 
to four given screws. They, therefore, lie on a cylindroid which we call P . 
In like manner, since Q is a system of the third order, the screws that can be 
selected from it, so as to be at the same time reciprocal to &>, will also form a 
cylindroid which we call Q . 

It is generally a possible and determinate problem to find, among the 
screws of a system of the third order, one screw which shall be reciprocal 
to every screw, on an arbitrary cylindroid. For, take three screws from the 
system reciprocal to the given system of the third order, and two screws on 
the given cylindroid. As a single screw can be found reciprocal to any five 
screws, the screw reciprocal to the five just mentioned will be the screw now 
desired. 

We apply this principle to obtain the screw 6, in the system P, which is 
reciprocal to the cylindroid Q . In like manner, we find the screw </>, in the 
system Q, which is reciprocal to the cylindroid P . 

From the construction it is evident that the three screws 0, <f>, and <w are 
all reciprocal to the two cylindroids P and Q . This is, of course, equivalent 
to the statement that 0, <j>, a> are all reciprocal to the screws of a system of 



307] THE GEOMETRICAL THEORY. 331 

the fourth order. It follows that, 6, </>, w must lie upon the same cylin- 
droid. Thus, 6, </> are the two screws required, and the problem has been 
solved. It is easily seen that there is only one such screw 0, and one such 
screw <>. 

Or we might have proceeded as follows : Take any three screws on P, 
and any three screws on Q. Then by a fundamental principle a wrench on <y 
can be decomposed into six component wrenches on these six screws. But 
the three component wrenches on P will compound into a single wrench on 
some screw 6 belonging to P. The three component wrenches on Q will 
compound into a single wrench on some screw </> belonging to Q. Thus the 
original wrench on <w may be completely replaced by single wrenches on 6 
and <f>. But this proves that 6, </>, and w are co-cylindroidal. 

In the special case of this theorem which we are now to use one of the 
systems of the third order assumes an extreme type. It consists simply of 
all possible screws of infinite pitch. The theorem just proved asserts that 
in this case a twist velocity about any screw <w can always be replaced by a 
twist velocity about some one screw belonging to any given system of the 
third order P, together with a suitable velocity of translation. 

In the problem before us we know three corresponding pairs of impulsive 
screws and instantaneous screws (rj, 2), (, /3), (, 7), and we seek the impul 
sive screw corresponding to some fourth instantaneous screw 8. 

It should be noticed that the data are sufficient but not redundant. We 
have seen how a knowledge of two pairs of corresponding impulsive screws 
and instantaneous screws provided eight of the coordinates of the rigid 
body. The additional pair of corresponding screws only bring one further 
co-ordinate. For, though the knowledge of 7 appropriate to a given f 
might seem five data, yet it must be remembered that the two pairs (??, a) 
and (f, 7) must fulfil the two fundamental geometrical conditions, and so 
must also the two pairs (, /3) and (, 7) ; thus, as 7 has to comply with 
four conditions, it really only conveys a single additional coordinate, which, 
added to the eight previously given, make the nine which are required for 
the rigid body. We should therefore expect that the knowledge of three 
corresponding pairs must suffice for the determination of every other pair. 

Let the unit twist velocity about 8 be resolved by the principles ex 
plained in this section into a twist velocity on some screw S belonging to 
a, /3, 7, and into a velocity of translation on a screw x of infinite pitch. 

We have already seen that the impulsive screw corresponding to 8 must 
lie on the system of the third order defined by 77, , and and that it 
is definitely determined. Let us denote by -^ this known impulsive screw 
which would make the body commence to twist about B . 



332 THE THEORY OF SCREWS. [307- 

Let the centre of gravity be constructed as in the last section ; then an 
impulsive force through the centre of gravity will produce the velocity of 
translation on 8j. Let us denote by % the screw of zero pitch on which this 
force lies. 

We thus have % as the impulsive screw corresponding to the instan 
taneous screw S^ while i/r is the impulsive screw corresponding to the 
instantaneous screw S . 

Draw" now the cylindroids (%, i/r) and (8 1} S ). The first of these is 
the locus of the impulsive screws corresponding to the instantaneous 
screws on the second. As already explained, we can completely correlate 
the screws on two such cylindroids. We can, therefore, construct the 
impulsive screw on (^, ^) which corresponds to any instantaneous screw 
on (S 1; S ). It is, however, obvious, from the construction, that the original 
screw B lies on the cylindroid (S 1} S ). Hence we obtain the impulsive screw 
which corresponds to B as the instantaneous screw, and the problem has 
been solved. 

308. Another method. 

We might have proceeded otherwise as follows : From the three given 
pairs of impulsive screws and instantaneous screws rja, %J3, 7 we can find 
other pairs in various ways. For example, draw the cylindroids (a, #) 
and (, ); then select, by principles already explained, a screw B on the 
first cylindroid, and its correspondent 6 on the second. In like manner, 
from the cylindroids (a, 7) and (17, ), we can obtain another pair (</>, e). We 
have thus five pairs of correspondents, ??, /3, 7, 08, <f>e. Each of these 
will give a diameter of the momental ellipsoid, and the radius of gyration 
about that diameter. Thus we know the centre of the momental ellipsoid 
and five points on its surface. The ellipsoid can be drawn accordingly. 
Its three principal axes give the principal screws of inertia. All other 
pairs of correspondents can then be determined by a construction given 
later on (311). 

309. Unconstrained motion in system of second order. 

Suppose that a cylindroid be drawn through any two (not lying along 
the same principal axis) of the six principal screws of inertia of a free rigid 
body. If the body while at rest be struck by an impulsive wrench about 
any one of the screws of the cylindroid it will commence to move by 
twisting about a screw which also lies on the cylindroid. For the given 
impulsive wrench can be replaced by two component wrenches on any 
two screws of the cylindroid. We shall, accordingly, take the component 
wrenches of the given impulse on the two principal screws of inertia which, 
by hypothesis, are contained on the cylindroid. Each of those components 



309] THE GEOMETRICAL THEORY. 333 

will, by the property of a principal screw of inertia, produce an instantaneous 
twist velocity about the same screw. But the two twist velocities so 
generated can, of course, only compound into a single twist velocity on some 
other screw of the cylindroid. We have now to obtain the geometrical 
relations characteristic of the pairs of impulsive and instantaneous screws on 
such a cylindroid. 

In previous chapters we have discussed the relations between impulsive 
screws and instantaneous screws, when the movements of the body are 
confined, by geometrical constraint, to twists about the screws on a 
cylindroid. The problem now before us is a special case, for though the 
movements are no other than twists about the screws on a cylindroid, yet 
this restriction, in the present case, is not the result of constraint. It arises 
from the fact that two of the six principal screws of inertia of the rigid 
body lie on the cylindroid, while the impulsive wrench is, by hypothesis, 
limited to the same surface. 

To study the question we shall make use of the circular representation 
of the cylindroid, 50. We have there shown how, when the several screws 
on the cylindroid are represented by points on the circumference of a circle, 
various dynamical problems can be solved with simplicity and convenience. 
For example, when the impulsive screw is represented on the circle by 
one point, and the instantaneous screw by another, we have seen how 
these points are connected by geometrical construction ( 140). 

In the case of the unconstrained body, which is that now before us, it is 
known that, whenever the pitch of an instantaneous screw is zero, the corre 
sponding impulsive screw must be at right angles thereto ( 301). 

In the circular representation, the angle between any two screws is 
equal to the angle subtended in the representative circle by the chord 
whose extremities are the representatives of the two screws. Two screws, 
at right angles, are consequently represented by the extremities of a 
diameter of the representative circle. If, therefore, we take A, B, two 
points on the circle, to represent the two screws of zero pitch, then the two 
points, P and Q, diametrically opposite to them, are the points indicating 
the corresponding impulsive screws. It is plain from 287 that AQ and BP 
must intersect in the homographic axis, and hence the homographic axis 
is parallel to AQ and EP, and as it must contain the pole of AB it follows 
that the homographic axis XY must be the diameter perpendicular to AB. 

The two principal screws of the cylindroid X and Y are, in this case, the 
principal screws of inertia. Each of them, when regarded as an impulsive 
screw, coincides with its corresponding instantaneous screw. The diameter 
XY bisects the angle between AP and BQ. 



334 THE THEORY OF SCREWS. [309, 

It is shown ( 137) that the points which represent the instantaneous 
screws, and the points which represent the corresponding impulsive screws, 
form two homographic systems. A well-known geometrical principle asserts 
( 146), that if each point on a circle be joined to its homographic corre 
spondent, the chord will envelop a conic which has double contact with the 
circle. It is easily seen that, in the present case, the conic must be the 
hyperbola which touches the circle at the ends of the diameter XY, and 
has the rays AP and BQ for its asymptotes. The hyperbola is completely 
defined by these conditions, so that the pairs of correspondents are uniquely 
determined. 

Every tangent, 1ST, to this hyperbola will cut the circle in two points.. 
I and S, such that 8 is the point corresponding to the impulsive screw, and 
/ is the point which marks out the instantaneous screw. We thus obtain 
a concise geometrical theory of the connexion between the pairs of cor 
responding impulsive screws and instantaneous screws on a cylindroid which 
contains two of the principal screws of inertia of a free rigid body. 

For completeness, it may be necessary to solve the same problem Avhen 
the cylindroid is defined by two principal screws of inertia lying along the 
same principal axis of the rigid body. It is easy to see that if, on the 
principal axis, whose radius of gyration was a, there lay any instantaneous 
screw whose pitch was p a , then the corresponding impulsive screw would 
be also on the same axis, and its pitch would be p n where p n x p a = a?. 

310. Analogous Problem in a Three-system. 

Let us now take the case of the system of screws of the third order, 
which contains three of the six principal screws of inertia of a free rigid 
body. 

Any impulsive wrench, which acts on a screw of a system of the third 
order, can be decomposed into wrenches on any three screws of that system, 
and consequently, on the three principal screws of inertia, which in the 
present case the three-system has been made to contain. Each of these 
component wrenches will, from the property of a principal screw of inertia, 
generate an initial twist velocity of motion around the same screw. The 
three twist velocities, thence arising, can be compounded into a single twist 
velocity about some other screw of the system. We desire to obtain the 
geometrical relation between each such resulting instantaneous screw and 
the corresponding impulsive screw. 

It has been explained in Chap. XV. how the several points in a plane 
are in correspondence with the several screws which constitute a system 
of the third order. It was further shown, that if by the imposition of 
geometrical constraints, the freedom of a rigid body was limited to twisting 



310] THE GEOMETRICAL THEORY. 335 

about the several screws of the system of the third order, a geometrical 
construction could be obtained for determining the point corresponding to 
any instantaneous screw, when the point corresponding to the appropriate 
impulsive screw was known. 

We have now to introduce the simplification of the problem, which 
results when three of the principal screws of inertia of the body belong 
to the system. But a word of caution, against a possible misunderstanding, 
is first necessary. It is of course a fundamental principle, that when a 
rigid system has freedom of the ?ith order, there will always be, in the 
system of screws expressing that freedom, n screws such that an im 
pulsive wrench administered on any one of those screws will immediately 
make the body begin to move by twisting about the same screw. These 
are the n principal screws of inertia. 

But in the case immediately under consideration the rigid body is sup 
posed to be free, and it has, therefore, six principal screws of inertia. The 
system of the third order, at present before us, is one which contains three 
of these principal screws of inertia of the free body. Such a system of 
screws possesses the property, that an impulsive wrench on any screw 
belonging to it will set the body twisting about a screw which also belongs 
to the same system. This is the case even though, in the total absence of 
constraints, there is no kinematical difficulty about the body twisting about 
any screw whatever. 

As there are no constraints, we know that each instantaneous screw, of 
zero pitch, must be at right angles to the corresponding impulsive screw 
( 301). This condition will enable us to adjust the particular homography 
in the plane wherein each pair of correspondents represents an impulsive 
screw and the appropriate instantaneous screw. 

The conic, which is the locus of points corresponding to the screws of a 
given pitch p, has as its equation ( 204) 

3 *-p(6? + 6* + 6> 3 2 ) = 0. 



The families of conies corresponding to the various values of p have a 
common self-conjugate triangle. The vertices of that triangle correspond to 
the three principal screws of inertia. 

The three points just found are the double points of the homography 
which correlate the points representing the impulsive screws with those 
representing the instantaneous screws. Let us take the two conies of the 
system, corresponding to p = and p = oo . They are 

3 2 = ........................... (i), 

3 2 = ........................... (ii). 



336 THE THEORY OF SCREWS. [310, 

Two conjugate points to conic (i) denote two reciprocal screws. Two con 
jugate points to conic (ii) denote two screws at right angles. 

Let A be any point representing an instantaneous screw. Take the 
polar of A, with respect to conic (i). Let P be the pole of this ray, with 
respect to conic (ii). 

Then P will correspond to the impulsive screw, while A corresponds to 
the appropriate instantaneous screw. For this is clearly a homography of 
which A and P are two correspondents. Further, the double points of this 
homography are the vertices of the common conjugate triangle to conies (i) 
and (ii). If A lie on (i), then its polar is the tangent to (i) ; and as every 
point on this polar will be conjugate to P, with respect to conic (ii), it 
follows that A and P are conjugate, with respect to (ii) that is, A and P 
are correspondents of a pair of screws at right angles. As the pitch of the 
screw, corresponding to A, is zero, we have thus obtained the solution of 
our problem. 

311. Fundamental Problem with Free Body. 

We now give the geometrical solution of the problem so fundamental 
in this present theory which may be thus stated : 

A perfectly free body at rest is struck by an impulsive wrench upon a 
given screw. It is required to construct the instantaneous screw about which 
the body will commence to twist. 

The rigid body being given, its three principal axes are to be drawn 
through its centre of gravity. The radii of gyration a, b, c about these 
axes are to be found. On the first principal axis two screws of pitches + a 
and a respectively are to be placed. Similarly screws of pitches + b, b, 
and +c, c are to be placed on the other two principal axes. These are, of 
course, the six principal screws of inertia: call them A 0> A l} A. 2 , A 3 , A 4> A 5 . 
We then draw the five cylindroids 

AoA 1} A A 2 , A A 3> A Q At, A A B . 

It is always possible to find one screw on a cylindroid reciprocal to any 
given screw. In certain cases, however, of a special nature, more than a 
single screw can be so found. Under such circumstances the present 
process is inapplicable, but the exceptional instances will be dealt with 
presently. 

Choose on the cylindroid A A l a screw 1 which is reciprocal to the 
given impulsive screw ?;, which is, of course, supposed to lie anywhere and 
be of any pitch. 

In like manner, choose on the other four cylindroids screws 2 , 3 , # 4 , 6 6 , 
respectively, all of which are also reciprocal to 77. 



311] THE GEOMETRICAL THEORY. 337 

Let us now think of 9 l as an instantaneous screw ; it lies on the cylindroid 
A A lt and this cylindroid contains two principal screws of inertia. It follows 
from 309 that the corresponding impulsive screw fa lies on the same cylin 
droid. That screw fa can be determined by the construction there given. 
In like manner we construct on the other four cylindroids the screws <j> 2 , fa, 
fa, fa, which are the impulsive screws corresponding respectively to 2 , 3 , 
# 4 , 6 , as instantaneous screws. 

Consider then the two pairs of corresponding impulsive screws and in 
stantaneous J screws (77, a) and (fa, ^). We have arranged. the construction 
so that 6 l is reciprocal to 77. Hence, by the fundamental principle so often 
employed, a and d l are conjugate screws of inertia, so that a must be reciprocal 
to fa. 

In like manner it can be proved that the instantaneous screw a for which 
we are in search must be reciprocal to fa, fa, fa, fa. We have thus dis 
covered five screws, fa, fa, fa, fa, fa, to each of which the required screw a 
must be reciprocal. But it is a fundamental point in the theory that the 
single screw reciprocal to five screws can be constructed geometrically ( 25). 
Hence a is determined, and the geometrical solution of the problem is 
complete. 

It remains to examine the failure in this construction which arises when 
any one or more of the five screws fa ... fa becomes indeterminate. This 
happens when 77 is reciprocal to two screws on the cylindroid in question. 
In this case 77 is reciprocal to every screw on the cylindroid. Any one of such 
screws might be taken as the corresponding fa and, of course, would have 
been also indefinite, and a could not have been found. In this case 77 would 
have been reciprocal to the two principal screws of inertia, suppose A , A l 
which the cylindroid contained. Of course still more indeterminateness 
would arise if 77 had been also reciprocal to other screws of the series A , A l} 
A a , A 3 , A 4 , A 5 . No screw could, however, be reciprocal to all of them. If?; 
had been reciprocal to five, namely, A l} A a , A s , A,, A 5 , then 77 could be no 
screw other than A , because the six principal screws of inertia are co- 
reciprocal ; 77 would then be its own instantaneous screw, and the problem 
would be solved. 

We may therefore, under the most unfavourable conditions, take 77 to be 
reciprocal to four of the principal screws of inertia A , A lt A a , A 3> but not to 
A 4 or .-1 5 . We now draw the five cylindroids, A A t , A,A 4 , A 2 A 4 , A 3 A 4 , A A 5 . 
We know that 77 is reciprocal to no more than a single screw on each 
cylindroid. We therefore proceed to the construction as before, first finding 
0, ... 5 , one on each cylindroid ; then deducing fa ... fa, and thus ultimately 
obtaining a. 

Thus the general problem has been solved. 

B. 22 



338 THE THEORY OF SCREWS. [312- 

312. Freedom of the First or Second Order. 

If the rigid body have only a single degree of freedom, then the only 
movements of which it is capable are those of twisting to and fro on a 
single screw a. If the impulsive wrench 77 which acted upon the body 
happened to be reciprocal to a, then no movement would result. The forces 
would be destroyed by the reactions of the constraints. In general, of course, 
the impulsive screw 77 will not be reciprocal to a. A twisting motion about 
a will therefore be the result. All that can be said of the instantaneous screw 
is that it can be no possible screw but a. 

In the next case the body has two degrees of freedom which, as usual, we 
consider to be of the most general type. It is required to obtain a con 
struction for the instantaneous screw a about which a body will commence 
to twist in consequence of an impulsive wrench 77. 

The peculiarity of the problem when the notion of constraint is introduced 
depends on the circumstance that, though the impulsive screw may be 
situated anywhere and be of any pitch, yet that as the body is restrained to 
only two degrees of freedom, it can only move by twisting about one of the 
screws on a certain cylindroid. We are, therefore, to search for the in 
stantaneous screw on the cylindroid expressing the freedom. 

Let A be the given cylindroid. Let B be the system of screws of the 
fourth order reciprocal to that cylindroid. If the body had been free it would 
have been possible to determine, in the manner explained in the last section, 
the impulsive screw corresponding to each screw on the cylindroid A. Let 
us suppose that these impulsive screws are constructed. They will all lie on 
a cylindroid which we denote as P. In fact, if any two of such screws had 
been found, P would of course have been denned by drawing the cylindroid 
through those two screws. 

Let Q be the system of screws of the fourth order which is reciprocal to 
P. Select from Q the system of the third order Q t which is reciprocal to 77. 
We can then find one screw ^ which is reciprocal to the system of the fifth 
order formed from A and Qj. It is plain that -^ must belong to B, as this 
contains every screw reciprocal to A. 

Take also the one screw on the cylindroid A which is reciprocal to 77, and 
find the one screw rj 2 on the cylindroid P which is reciprocal to this screw 
on A. 

Since 77, 77! and ?7 2 are all reciprocal to the system of the fourth order 
formed by A l and Q 1} it follows that 77, 77^ and 773 must all lie on the same 
cylindroid. We can therefore resolve the original wrench on 77 into two com 
ponent wrenches on y^ and 772. 



314] THE GEOMETRICAL THEORY. 339 

But it is of the essence of the theory that the reactions of the constraints 
by which the motion of the body is limited to twists about screws on the 
cylindroid A must be wrenches on the reciprocal system B. So far, therefore, 
as the body thus constrained is concerned, the reactions of the constraints 
will neutralize the wrench on TJ I . Thus the wrench on r} 2 is the only part of 
the impulsive wrench which need be considered. 

But we already know from the construction that an impulsive wrench on^ 
will produce an instantaneous twist velocity about a determined screw a on 
A. Thus we have found the initial movement, and the investigation is geo 
metrically complete. 

313. Freedom of the Third Order. 

We next consider the case in which a rigid body has freedom of the third 
order. We require, as before, to find a geometrical construction for the in 
stantaneous screw a corresponding to a given impulsive screw 77. 

Let A be the system of screws of the third order about which the body 
is free to twist. Let B be the system of screws of the third order reciprocal 
to A. We must first construct the system of the third order P which 
consists of the impulsive screws that would have made the body, if perfectly 
free, twist about the several screws of A. 

As already explained ( 307) we can, in one way, but only in one way, 
resolve the original wrench on 77 into wrenches 7^ on B, and 772 on P. The 
former is destroyed by the reactions of the constants. The latter gives rise 
to a twist velocity about a determinate screw on A. Thus the problem has 
been solved. 

314. General Case. 

We can obviously extend a similar line of reasoning to the cases where 
the body had freedom of the remaining degrees. It will, however, be as simple 
to write the general case at once. 

Let A be a system of screws of the ?ith order, about which a body is free 
to twist, any other movements being prevented by constraints. If the body 
receive an impulsive wrench, on any screw 77, it is required to determine the 
instantaneous screw, of course belonging to A, about which the body will 
commence to twist. 

Let B be the system of screws of the (6 - w)th order, reciprocal to A. 
The wrenches arising from the reaction of the constraints must, of course, be 
situated on the screws of the system B. 

Let P be the system of screws of the nth order, which, in case the body 

222 



340 THE THEORY OF SCREWS. [314- 

had been free, would have been the impulsive screws, corresponding to the 
instantaneous screws belonging to A. 

Let Q be the system of screw of the (6 - n)th order, which are reciprocal 
to P. 

Take from A the system of the (n- l)th order, reciprocal to 77, and call 
it A,. 

Take from Q the system of the (5 ?i)th order, reciprocal to 77, and call 
it Q,. 

As A is of the nth order, and Q x of the (5 - w)th, they together define a 
system of the fifth order. Let ^ be the single screw, reciprocal to this system 
of the fifth order. 

As Jj is of the (n-l)th order, and Q is of the (6 - w)th order, they 
together define a system of the fifth order. Let r} 2 be the single screw, 
reciprocal to this system of the fifth order. 

77! is reciprocal to A l of the (n l)th order, because J.j forms part of A. 
It is reciprocal to Qj of the (5 n)th order, because it was so made by con 
struction. Thus 77! is reciprocal to both A l and Q lt that is to a system of the 
fourth order. 

In like manner, it can also be shown that rj 2 is reciprocal to both A^ 
and Q lt 

A l and Q 1 were originally chosen so as to be reciprocal to 77. It thus 
appears that the three screws, rj, tj^, 772, are all reciprocal to the same system 
of the fourth order. They are, therefore, co-cylindroidal. 

The initial wrench on 77 can therefore be adequately replaced by two 
components on ^ and ?7 2 . The former of these is destroyed by the reaction 
of the constraints. The latter gives rise to an initial movement on a 
determined screw of A. Thus, the most general problem of the effect of 
an impulsive wrench on a constrained rigid body has been solved geo 
metrically. 

315. Freedom of the Fifth Order. 

The special case, where the rigid body has freedom of the fifth order, may 
be viewed as follows. 

Let p be the screw reciprocal to the screw system of the fifth order, about 
which the body is free to twist. 

Let A, be the instantaneous screw, determined in the way already explained, 
about which the body, had it been free, would have commenced to twist in 
consequence of an impulsive wrench on p. 



316] THE GEOMETRICAL THEORY. 341 

Let 77 be any screw on which an impulsive wrench is imparted, and let a 
be the corresponding instantaneous screw, about which the body would have 
begun to twist had it been free. 

Draw the cylindroid through a and X, and choose on this cylindroid the 
screw /JL, which is reciprocal to p. 

Then ^ is the instantaneous screw about which the body commences to 
twist, in consequence of the impulsive wrench on 77. 

For (77, p) is a cylindroid of impulsive screws, and (a, X) are the corre 
sponding instantaneous screws. As p is reciprocal to p, it belongs to the 
system of the fifth order. The corresponding impulsive screw must lie on 
(77, p). The actual instantaneous motion could therefore have been produced 
by impulsive wrenches on 77 and p. The latter would, however, be neutralized 
by the reactions of the constraints. We therefore find that 77 is the impulsive 
screw, corresponding to a as the instantaneous screw. 

316. Principal Screws of Inertia of Constrained Body. 

There is no more important theorem in this part of the Theory of Screws 
than that which affirms that for a rigid body, with n degrees of freedom, 
there are n screws, such that if the body when quiescent receives an 
impulsive wrench about one of such screws, it will immediately commence 
to move by twisting about the same screw. 

We shall show how the principles, already explained, will enable us to 
construct these screws. 

We commence with the case in which the body has two degrees of 
freedom. We take three screws, 77, , , arbitrarily selected on the 
cylindroid, which expresses the freedom of the body. We can then de 
termine, by the preceding investigation, the three instantaneous screws, 
, ft, 7, on the same cylindroid, which correspond, respectively, to the 
impulsive screws. Of course, if 77 happened to coincide with a, or with ft, 
or f with 7, one of the principal screws of inertia would have been found. 
But, in general, such pairs will not coincide. We have to show how, from 
the knowledge of three such pairs, in general, the two principal screws of 
inertia can be found. 

We employ the circular representation of the points on the cylindroid, 
as explained in 50. The impulsive screws are represented by one system 
of points, the corresponding instantaneous screws are represented by another 
system of points. It is an essential principle, that the two systems of points, 
so related, are homographic. The discovery of the principal screws of inertia 
is thus reduced to the well-known problem of the discovery of the double 
points of two homographic systems on a circle. 



342 THE THEORY OF SCREWS. [316, 

The simplest method of solving this problem is that already given in 
139, in which we regard the six points, suitably arranged, as the vertices 
of a hexagon ; then the Pascal line of the hexagon intersects the circle in 
two points which are the points corresponding to the principal screws of 
inertia. 

317. Third and Higher Systems. 

We next investigate the principal screws of inertia of a body which has 
three degrees of freedom. We have first, by the principles already ex 
plained, to discover four pairs of correspondents. When four such pairs 
are known, the principal screws of inertia can be constructed. Perhaps 
the best method of doing so is to utilize the plane correspondence, as 
explained in Chap. xv. The corresponding systems of impulsive screws 
and instantaneous screws, in the system of the third order, are then repre 
sented by the homographic systems of points in the plane. When four pairs 
of such correspondents are known, we can construct as many additional pairs 
as may be desired. 

Let a, 0, 7, 8 be four points in the plane, and let 77, , f, be the 
points corresponding, so that ?/ represents the impulsive screw, and a the 
instantaneous screw, and similarly for the other pairs. Let it be required 
to find the impulsive screw <, which corresponds to any fifth instantaneous 
screw e. Since anharmonic ratios are the same in two corresponding figures, 
we have 

a (0, 7, 8, 6) = ,K ?, 0, (/>), 
thus we get one ray r)<f>, which contains <. We have also 

(, 7, , <0 = f07, C, 0. <), 
which gives a second ray <, containing </>, and thus <f> is known. 

A construction for the double points of two homographic systems of 
points in the same plane is as follows: 

Let and be a pair of corresponding points. Then each ray 
through will have, as its correspondent, a ray through . The locus of 
the intersection of these rays will be a conic S. This conic S must pass 
through the three double points, and also through and . 

Draw the conic 8 , which is the locus of the points in the second system 
corresponding to the points on S, regarded as in the first system. Then 
since lies on S, we must have on S . But S must also pass through 
the three double points. is one of the four intersections of S and S , and 
the three others are the sought double points. Thus the double points are 
constructed. And in this manner we obtain the three principal screws of 
inertia in the case of the system of the third order. 



317] THE GEOMETRICAL THEORY. 343 

If a rigid body be free to move about screws of any system of the 
fourth order, we may determine its four principal screws of inertia as 
follows. 

We correlate the several screws of the system of the fourth order with 
the points in space. The points representing impulsive screws will be 
a system homographic with the points representing the corresponding 
instantaneous screws. If we have five pairs of correspondents (a, 77), 
(@> )> (% 0> (S, &), (e, <p), in such a homography we can at once determine 
the correspondent ^ to any other screw X. 

Draw a pencil of four planes through a, /3, and the four points j, B, e, X, 
respectively. 

Draw also a pencil of four planes through ij, , and the four points 
, 0, (j), T/T, respectively. These two pencils will be equianharmonic. Thus 
we discover one plane which contains i/r. In like manner we can draw 
a pencil of planes through a, y, and /3, 8, e, X, respectively, and the equi 
anharmonic pencil through 77, , and , 6, <f>, ^, respectively. Thus we 
obtain a second plane which passes through ty. A third plane may be 
found by drawing the pencil of four planes through a, S, and the four points 
/3, y, e, X, respectively, and then constructing the equianharmonic pencil 
through 77, 6, and , , 0, \^, respectively. From the intersection of these 
three planes, ^r is known. 

In the case of two homographic systems in three dimensions, there are, 
of course, four double points. These may be determined as follows. 
Let and be two corresponding rays. Then any plane through will 
have, as its correspondent, a plane through . It is easily seen that these 
planes intersect on a ray which has for its locus a quadric surface S, of which 
and are also generators. This quadric must pass through the four 
double points. 

Let 8 be the quadric surface which contains all the points in the 
second system corresponding to the points of S regarded as the first system. 
Then will lie on 8 , and the rest of the intersection of 8 and S will be 
a twisted cubic G, which passes through the four double points. 

Take any point P on C, and draw any plane through P. Then every 
ray of the first system of the pencil through P in this plane will have 
as its correspondent in the second system the ray in some other plane 
pencil L. One, at least, of the rays in the pencil L will cut the cubic C. 
Call this ray X , and draw its correspondent X in the first system passing 
through P. 

We thus have a pair of corresponding rays X and X , each of which 
intersects the twisted cubic C. 



344 THE THEORY OF SCREWS. [317, 

Draw pairs of corresponding planes through X and X . The locus of 
their intersection will be a quadric S", which also contains the four double 
points. 

S" and C, being of the second and the third order respectively, will 
intersect in six points. Two of these are on X and X , and are thus dis 
tinguished. The four remaining intersections will be the required double 
points, and thus the problem has been solved. 

These double points correspond to the principal screws of inertia, which 
are accordingly determined. 

In the case of freedom of the fifth order, the geometrical analogies which 
have hitherto sufficed are not available. We have to fall back on the 
general fact that the impulsive screws and the corresponding instantaneous 
screws form two homographic systems. There are five double screws be 
longing to this homography. These are the principal screws of inertia. 

318. Correlation of Two Systems of the Third Order. 

It being given that a certain system of screws of the third order, P, is 
the locus of impulsive screws corresponding to another given system of the 
third order, A, as instantaneous screws, it is required to correlate the corre 
sponding pairs on the two systems. 

We have already had frequent occasion to use the result demonstrated 
in 293, namely, that when two impulsive and instantaneous cylindroids 
were known, we could arrange the several screws in corresponding pairs 
without any further information as to the rigid body. We have now to 
demonstrate that when we are given an impulsive system of the third 
order, and the corresponding instantaneous system, there can also be a 
similar adjustment of the corresponding pairs. 

It has first to be shown, that the proposed problem is a definite one. 
The data before us are sufficient to discriminate the several pairs of screws, 
that is to say, the data are sufficient to point out in one system the corre 
spondent to any specified screw in the other system. We have also to 
show that there is no ambiguity in the solution. There is only one rigid 
body ( 293) which will comply with the condition, and it is not possible that 
there could be more than one arrangement of corresponding pairs. 

Let a, /3, 7 be three instantaneous screws from A, and let their corre 
sponding impulsive screws be 77, , in P. In the choice of a screw from 
a system of the third order there are two disposable quantities, so that, 
in the selection of three correspondents in P, to three given screws in A, 
there would be, in general, six disposable coordinates. But the fact that 
a, ?; and @, g are two pairs of correspondents necessitates, as we know, 



318J THE GEOMETRICAL THEORY. 345 

the fulfilment of two identical conditions among their coordinates. As 
there are three pairs of correspondents, we see at once that there are six 
equations to be fulfilled. These are exactly the number required for the 
determination of rj, , , in the system P. 

To the same conclusion we might have been conducted by a different 
line of reasoning. It is known that, for the complete specification of a 
system of the third order, nine co-ordinates are necessary ( 75). This is 
the same number as is required for the specification of a rigid body. If, 
therefore, we are given that P, a system of the third order, is the collection 
of impulsive screws, corresponding to the instantaneous screws in the 
system A, we are given nine data towards the determination of a rigid 
body, for which A and P would possess the desired relation. It therefore 
follows that we have nine equations, while the rigid body involves nine 
unknowns. Thus we are led to expect that the number of bodies, for which 
the arrangement would be possible, is finite. When such a body is de 
termined, then of course the correlation of the screws on the two systems 
is immediately accomplished. It thus appears that the general problem 
of correlating the screws on any two given systems of the third order, 
A and P, into possible pairs of impulsive screws and instantaneous screws, 
ought not to admit of more than a finite number of solutions. 

We are now to prove that this finite number of solutions cannot be 
different from unity. 

For, let us suppose that a screw X, belonging to A, had two screws 
6 and <f>, as possible correspondents in P. This could, of course, in no case 
be possible with the same rigid body. We shall show that it could not 
even be possible with two rigid bodies, M { and M 2 . For, if two bodies could 
do what is suggested, then it can be shown that there are a singly infinite 
number of possible bodies, each of which would afford a different solution of 
the problem. 

We could design a rigid body in the following manner : 

Increase the density of every element of Mj in the ratio of p^ : 1, and 
call the new mass M. 

Increase the density of every element of M 2 in the ratio of p 2 : 1, and 
call the new mass M 2 . 

Let the two bodies, so altered, be conceived bound rigidly together by 
bonds which are regarded as imponderable. 

Let ty be any screw lying on the cylindroid (6, <j>). Then the impulsive 
wrench of intensity, ty " on i/r, may be decomposed into components 

, sin (^r-<J>) , , sin (6 - ^) 

V /d ,( on 9, and iK -^ ^ *g on <f>. 
sin (6 - </>) sm(0 - <) 



346 THE THEORY OF SCREWS. [318, 

If the former had been applied to M^ it would have generated about a 
( 280) a twist velocity represented by 

, sin (A/T - <f>) J^ cos(0a) 
r sin (6 r -</>) M t p a 

If the latter had been applied to the body M 2 , it would have generated a 
twist velocity about the same screw, a, equal to 

, sin (6 ^r) J^ cos ((fta) 
* sin (0-0) J/7 "~ Pa 

Suppose that these two twist velocities are equal, it is plain that the original 
wrench on ty would, if it had been applied to the composite rigid body, 
produce a twisting motion about a. The condition is 

sin (^r </>) _ sin (6 -fy) 
MI cos ((pa) ~ M 3 cos (0a) 

We thus obtain tan i/r in terms of MI : M a . As the structure of the 
composite body changes by alterations of the relative values of p l and p. z , 
so will T/T move over the various screws of the cylindroid (6, <p). 

This result shows that, if three screws, a, ft, 7 be given, then the possible 
impulsive screws, 77, , which shall respectively correspond to a, /3, 7 in a 
given system of the third order P, are uniquely determined. 

For, suppose that a second group of screws, 77 , % > > could also be deter 
mined which fulfilled the same property. We have shown how another 
rigid body could be constructed so that another screw, ^, could be found on 
the cylindroid (77, 77 ), such that an impulse thereon given would make the 
body twist about a. It is plain that, for this body also, the impulsive 
wrench, corresponding to ft, would be some screw on the cylindroid (f, ). 
But all screws on this cylindroid belong to the system P. In like manner, 
the instantaneous screw 7 would correspond for the composite body to some 
screw on the cylindroid (, ") Hence it follows that, for each different 
value of the ratio p^ : p 2 , we would have a different set of impulsive screws 
for the instantaneous screws a, ft, 7. We thus find that, if there were 
more than one set of such impulsive screws to be found in the system P, 
there would be an infinite number of such sets. But we have already 
shown that the number of sets must be finite. Hence there can only be 
one set of screws, 77, , , in the system P, which could be impulsive screws 
corresponding to the instantaneous screws, a, ft, 7. We are thus led to the 
following important theorem, which will be otherwise proved in the next 
chapter. 

Given any two systems of screws of the third order. It is generally pos 
sible, in one way, but only in one, to design, and place in a particular position a 



319] THE GEOMETRICAL THEORY. 347 

rigid body such that, if that body, while at rest and unconstrained, receive an 
impulsive wrench about any screw of the first system, the instantaneous move 
ment will be a twist about a screw of the second system. 

The two systems of corresponding impulsive and instantaneous screws 
on the two systems of the third order, form two homographic systems. 
There are, of course, infinite varieties in the possible homographic cor 
respondences between the screws of two systems of the third order. The 
number of such correspondences is just so many as the number of possible 
homographic correspondences of points in two planes. There is, however, 
only one correspondence which will fulfil the peculiar requirements when 
one of the systems expresses the instantaneous screws, and the other the 
impulsive screws severally corresponding thereto. 

If we are given one pair of corresponding impulsive and instantaneous 
screws, the body is not by such data fully determined. We are only given 
five coordinates, and four more remain, therefore, unknown. If we are 
given two corresponding impulsive cylindroids and instantaneous cylindroids, 
the body is still not completely specified. We have seen how eight of its 
coordinates are determined, but there is still one remaining indeterminate. 
If we are given a system of the fourth order of impulsive screws, and the 
corresponding system of the fourth order of instantaneous screws, the body, 
as in the other cases, remaining perfectly free, there are also, as we shall see 
in the next section, a singly infinite number of rigid bodies which fulfil the 
necessary conditions. In like manner, it will appear that, if we are given a 
system of the fifth order consisting of impulsive screws, and a corresponding 
system of the fifth order consisting of instantaneous screws, the body has 
really as much indeterminateness as if we had only been given a single 
pair of corresponding screws. 

But the case of two systems of the third order is exceptional, in that 
when it is known that one of these is the locus of the instantaneous screws, 
which correspond to the screws of the other system regarded as impulsive 
screws, the rigid body for which this state of things is possible is completely 
and uniquely specified as to each and every one of its nine coordinates. 

319. A Property of Reciprocal Screw Systems. 

Given a system of the fourth order A and another system of the fourth 
order P. If it be known that the latter is the locus of the screws on which 
must lie the impulsive wrenches which would, if applied to a free rigid 
body, cause instantaneous twist velocities about the several screws on A, 
let us consider what can be inferred as to the rigid body from this fact alone. 

Let A be the cylmdroid which is composed of the screws reciprocal to A. 



348 THE THEORY OF SCREWS. [319, 

Let P be the cylindroid which is composed of the screws reciprocal to P. 
Let P l} P 2 , P 3 , P 4 be any four impulsive screws on P. Let A 1} A 2 , A 3 , A 4 
be the four corresponding instantaneous screws on A. 

Take any screw a. on the cylindroid P . Let 77 be the corresponding 
impulsive screw. Since a is reciprocal to all the screws on P it must be 
reciprocal to Pj. It follows from the fundamental property of conjugate 
screws of inertia, that 77 must be reciprocal to A^. In like manner we can 
show that 77 is reciprocal to A 2 , A z , and A t . It follows that 77 is reciprocal 
to the whole system A, and therefore must be contained in the reciprocal 
cylindroid A. Hence we obtain the following remarkable result, which is 
obviously generally true, though our proof has been enunciated for the 
system of the fourth order only. 

Let P and A be any two systems of screws of the nth order, and P and A 
their respective reciprocal systems of the (ft~n)th order. If P be the collec 
tion of impulsive screws corresponding severally to the screws of A as the 
instantaneous screws for a certain free rigid body ; then, for the same free 
rigid body A will be the collection of impulsive screws which correspond to 
the screws of P as instantaneous screws. 

320. Systems of the Fourth Order. 

Thus we see that when we are given two systems of the fourth order P 
and A as correspondingly impulsive and instantaneous, we can immediately 
infer that, for the same rigid body, the screws on the cylindroid A are 
the impulsive screws corresponding to the instantaneous screws on the 
cylindroid P . 

We can now make use of that instructive theorem ( 293) which declares 
that when two given cylindroids are known to stand to each other in this 
peculiar relation, we are then able, without any further information, to mark 
out on the cylindroids the corresponding pairs of screws. We can then 
determine the centre of gravity of the rigid body on which the impulsive 
wrenches act. We can find a triad of conjugate diameters of the momental 
ellipsoid, and the radii of gyration about two of those diameters. Hence 
we have the following result : 

If it be given that a certain system of the fourth order is the locus of the 
impulsive screws corresponding to the instantaneous screws on another given 
system of the fourth order, the body being quite unconstrained, we can then 
determine the centre of gravity of the body, we can draw a triad of the 
conjugate diameters of its momental ellipsoid, and we can find ike radii of 
gyration about two of those diameters. 

There is still one undetermined element in the rigid body, namely, the 



320] THE GEOMETRICAL THEORY. 349 

radius of gyration about the remaining conjugate diameter. The data before 
us are not adequate to the removal of this indefiniteness. It must be 
remembered that t/he data in such a case are just so many but no more than 
suffice for the specification of the n-system A. The number of data neces 
sary to define an n-system is n (6 n). If, as in the present case, n = 4, the 
number of data is 8. We are thus one short in the number of data necessary 
to specify a rigid body. Thus we confirm the result previously obtained. 
We can assert that for any one of the singly infinite number of rigid bodies 
which fulfil the necessary conditions, the system A will be the locus of the 
instantaneous screws which correspond to the screws of the system P as 
impulsive screws. 

Though in the two cylindroids A and P we are able to establish the 
several pairs of correspondents quite definitely, yet we must not expect, with 
the data before us, to be able to correlate the pairs of screws in A and P 
definitely. If this could be done then the rigid body would be quite deter 
minate, which we know is not the case. There is, however, only a single 
indeterminate element left in the correlation of the screws in A with the 
screws of P. This we prove as follows : 

Let < be any screw of P on which an impulsive wrench is to act. Let 8 
be the instantaneous screw in A about which the movement commences. 
We shall now show that though 8 cannot be completely defined, in the 
absence of any further data, yet it can be shown to be limited to a certain 
cylindroid. 

Let G be the centre of gravity. Then we know that an impulsive force 
directed through G will generate a movement of translation in a direction 
parallel to the force. Such a movement may, of course, be regarded as a 
twist about a screw of infinite pitch. 

Draw through G a plane normal to </>. Any screws of infinite pitch in 
this plane will be reciprocal to <. It follows from the laws of conjugate screws 
of inertia that the impulsive forces in this plane, by which translations could 
be produced, must lie on screws of zero pitch which are reciprocal to 8. 
Take any two of such screws : then we know that 8 is reciprocal to these two 
screws and also to P . It follows that 8 is reciprocal to the screws of a 
determinate system of the fourth order, and therefore 8 must lie on a deter 
mined cylindroid. 

We may commence to establish the correspondence between P and A by 
choosing some arbitrary screw <f> on P, and then drawing the cylindroid on A, 
on which we know that the instantaneous screw corresponding to P must 
lie. Any screw on this cylindroid may be selected as the instantaneous 
screw which corresponds to </>. Once that screw 8 had been so chosen there 



350 THE THEORY OF SCREWS. [320- 

cau be no further ambiguity. The correspondent in A to every other screw 
in P is completely known. To show this it is only necessary to take two 
pairs from A and P and the pair just found. We have then three corre 
sponding pairs. It has been shown in 307 how the correspondence is 
completely determined in this case. 

Of course the fact that 8 may be any screw on a cylindroid is connected 
with the fact that in this case the rigid body had one indeterminate element. 
For each of the possible rigid bodies & would occupy a different position on 
its cylindroidal locus. 

321. Systems of the Fifth Order. 

It remains to consider the case where two screw systems of the fifth 
order are given, it being known that one of them P is the locus of the 
impulsive screws which correspond to the several screws of the other system 
A regarded as instantaneous screws. 

Let P be the screw reciprocal to P, and A the screw reciprocal to A. 
Then from the theorem of 319 it follows that an impulsive wrench on A 
would make the body commence to move by twisting about P . We thus 
know five of the coordinates of the rigid body. There remain four inde 
terminate elements. 

Hence we see that, when the only data are the two systems P and A , 
there is a fourfold infinity in the choice of the rigid body. There are conse 
quently four arbitrary elements in designing the correspondence between the 
several pairs of screws in the two systems. 

We may choose any two screws 77, , on P, and assume as their two corre 
spondents in A any two arbitrary screws a and ft, provided of course that 
the three pairs A , B , rj, a, and , ft fulfil the six necessary geometrical 
conditions ( 304). Two of these conditions are obviously already satisfied by 
the circumstance that A and P are the reciprocals to the systems A and P. 
This leaves four conditions to be fulfilled in the choice of a. and ft. As each 
of these belongs to a system of the fifth order there will be four coordinates 
required for its complete specification. Therefore there will be eight disposable 
quantities in the choice of o and ft. Four of these will be utilized in making 
them fulfil the geometrical conditions, so that four others may be arbitrarily 
selected. When these are chosen we have four coordinates of the rigid 
body which, added to the five data provided by A and P , completely define 
the rigid body. 

322. Summary. 

We may state the results of this discussion in the following manner : 
If we are given two systems of the first, or the second, or the third order 



323] THE GEOMETRICAL THEORY. 351 

of corresponding impulsive screws and instantaneous screws, all the corre 
sponding pairs are determined. There is no arbitrary element in the 
correspondence. There is no possible rigid body which would give any 
different correspondence. 

If we are given two systems of the fourth order of corresponding 
impulsive screws and instantaneous screws then the essential geometrical 
conditions ( 281), not here making any restriction necessary, we can select 
one pair of correspondents arbitrarily in the two systems, and find one rigid 
body to fulfil the requirements. 

If we are given two systems of the fifth order of corresponding impulsive 
screws and instantaneous screws then subject to the observance of the geo 
metrical conditions we can select two pairs of correspondents arbitrarily in 
the two systems, and find one rigid body to fulfil the requirements. 

If we are given two systems of the sixth order of corresponding impul 
sive screws and instantaneous screws then subject to the observance of 
the geometrical conditions we can select three pairs of correspondents 
arbitrarily in the two systems, and find one rigid body to fulfil the 
requirements. 

The last paragraph is, of course, only a different way of stating the results 
of 307. 

323. Two Rigid Bodies. 

We shall now examine the circumstances under which pairs of impulsive 
and instantaneous screws are common to two, or more, rigid bodies. The 
problem before us may, perhaps, be most clearly stated as follows : 

Let there be two rigid bodies, M and M . If M be struck by an impulsive 
wrench on a screw 6, it will commence to twist about some screw X. If M 
had been struck by an impulsive wrench on the same screw 6, the body would 
have commenced to twist about some screw /u,, which would of course be 
generally different from \. If 6 be supposed to occupy different positions 
in space (the bodies remaining unaltered), so will \ and p move into corre 
spondingly various positions. It is proposed to inquire whether, under 
any circumstances, 6 could be so placed that X and //, should coincide. In 
other words, whether both of the bodies, M and M , when struck with an 
impulsive wrench on 6, will respond by twisting about the same instantaneous 
screw. 

It is obvious, that there is at least one position in which 6 fulfils the 
required condition. Let 6^ G<> be the centres of gravity of M and M . Then 
a force along the ray G l 6r 2 , if applied either to M or to M , will do no more 
than produce a linear velocity of translation parallel thereto. Hence it 



352 THE THEORY OF SCREWS. [323 

follows, that a wrench on the screw of zero pitch, which lies on the ray 
G l G 2 , will have the same instantaneous screw whether that wrench be 
applied to M or to M . 

We have now to examine whether there can be any other pair of im 
pulsive and instantaneous screws in the same circumstances. Let us suppose 
that when 6 assumes a certain position 77, we have A, and /u, coalescing into the 
single screw a. 

We know that the centre of gravity lies in a plane through a, and the 
shortest distance between a and 77. We know, also, that d a =p a tan (a??), 
where d a is the distance of the centre of gravity from a. It therefore follows 
that a must be parallel to GiG 2 . We have, however, already had occasion 
( 303) to prove that, if p a be the radius of gyration of a body about a ray 
through its centre of gravity, parallel to a, 

2p a sr a ,, , 

pa 2 = -^f \ da ~ Pa- 

cos (377) 

Hence it appears that, for the required condition to be satisfied, each of the 
two bodies must have the same radius of gyration about the axis through its 
centre of gravity, which is parallel to a. Of course this will not generally be 
the case. It follows that, in general, there cannot be any such pair of 
impulsive screws and instantaneous screws, as has been supposed. Hence we 
have the following result : 

Two rigid bodies, with different centres of gravity, have, in general, no other 
common pair of impulsive screws and instantaneous screws than the screw, of 
zero pitch, on the ray joining the centres of gravity, and the screw of infinite 
pitch parallel thereto. 

We shall now consider what happens when the exceptional condition, just 
referred to, is fulfilled, that is, when the radius of gyration of the ray G 1 G 2 is 
the same for each of the bodies. 

In each of the momental ellipsoids about the centres of gravity of the 
two bodies, draw the plane conjugate to the ray G^G 2 . Let these planes 
intersect in a ray T. Suppose that an impulsive force, directed along T, be 
made to act on the body whose centre of gravity is O l . It is plain, from 
Poinsot s well-known theory, that the rotation produced by such an impulse 
will be about a ray parallel to G 1 G 2 . If this impulsive wrench had been 
applied to the body whose centre of gravity is G 2 , the instantaneous screw 
would also be parallel to G 1 G 2 . If we now introduce the condition that the 
radius of gyration of each of the bodies, about G 1 G 2 , is the same, it can be 
easily deduced that the two instantaneous screws are identical. Hence we 
see that T, regarded as an impulsive screw of zero pitch, will have the same 
instantaneous screw for each of the two bodies. 



323] THE GEOMETRICAL THEORY. 353 

If we regard T and G 1 G 2 as two screws of zero pitch, and draw the 
cylindroid through these two screws, then any impulsive wrench about a 
screw on this cylindroid will have the same instantaneous screw for either 
of the two bodies to which it is applied. 

For such a wrench may be decomposed into forces on T and on G 1 G 2 ] 
these will produce, in either body, a twist about a, and a translation parallel 
to a, respectively. We therefore obtain the following theorem : 

If two rigid bodies have different centres of gravity, G { and 6r 2 , and if 
their radii of gyration about the ray G^ z are equal, there is then a 
cylindroid of screws such that an impulsive wrench on any one of these 
screws will make either of the rigid bodies begin to twist about the same 
screw, and the instantaneous screws which correspond to the several screws 
on this cylindroid, all lie on the same ray Gfi^, but with infinitely varied 
pitch. 

It is to be remarked that under no other circumstances can any im 
pulsive screw, except the ray GiG z , with zero pitch, have the same instan 
taneous screw for each of the two bodies, so long as their centres of gravity 
are distinct. 

We might have demonstrated the theorem, above given, from the results 
of 303. We have there shown that, when an impulsive screw and the 
corresponding instantaneous screw are given, the rigid body must fulfil five 
conditions, the nature of which is fully explained. If we take two bodies 
which comply with these conditions, it appears that the ray through their 
centre of gravity is parallel to the instantaneous screw, and we also find 
that their radii of gyration must be equal about the straight line through 
their centres of gravity. 

If two rigid bodies have the same centre of gravity, then, of course, any 
ray through this point will be the seat of an impulsive wrench on a screw of 
zero pitch such that it generates a twist velocity on a screw of infinite pitch, 
parallel to the impulsive screw. This will be the case to whichever of the 
two bodies the force be applied. We have therefore a system of the third 
order (much specialized no doubt) of impulsive screws, each of which has the 
same instantaneous screw for each of the two bodies. In general there will 
be no other pairs of common impulsive and instantaneous screws beyond 
those indicated. 

Under certain circumstances, however, there will be other screws possess 
ing the same relation. 

We may suppose the two momental ellipsoids to be drawn about the 
common centre of gravity. These ellipsoids will, by a well-known property, 
possess one triad of common conjugate diameters. In general, of course, the 
B. 23 



354 THE THEORY OF SCREWS. [323 

radii of gyration of the two bodies, around any one of these three diameters, 
will not be equal. If, however, it should happen that the radius of gyration 
of one body be equal to that of the other body about one OX of these three 
common conjugate diameters, it can be shown that any screw, parallel to OX, 
whatever be its pitch regarded as an instantaneous screw, will have the same 
impulsive screw for either of the two bodies. 

If the radii of gyration about two of the common conjugate diameters 
were equal for the two bodies, it will then appear that any instantaneous 
screw which is parallel to the plane of the common conjugate diameters, will 
have the same impulsive screw for each of the two bodies. The corresponding 
impulsive screws belong to the system of the fifth order, which is defined by 
being reciprocal to a screw of zero pitch on the third of the three common 
conjugate diameters. 

Of course, if the radii of gyration coincide on this third diameter, then 
the two rigid bodies, regarded from our present point of view, would be 
identical. 



CHAPTER XXIII. 



VARIOUS EXERCISES. 

321 The Co-ordinates of a Rigid Body. 

We have already explained ( 302) how nine co-ordinates define a rigid 
body sufficiently for the present theory. One set of such co-ordinates with 
respect to any three rectangular axes may be obtained as follows. 

Let the element dm have the co-ordinates x, y, z, then causing the 
integrals to extend over the whole mass, we compute the nine quantities 

fxdm = Mx ; fydm = My ; fzdm = Mz ; 

jyzdm = Ml? ; fxzdm = M1 2 2 ; fay dm = M1 3 2 ; 

j(y* + z 2 ) dm = Mp, 2 ; JO 2 + z 2 ) dm = Mp 2 2 ; f(x 2 + f) dm = Mp 3 2 . 

The nine quantities #, y , z , I?, 1 2 2 , 1 3 2 , pf, p.?, p/ constitute an adequate 
system of co-ordinates of the rigid body. 

If 0J, 2 , ... 6 S be the canonical co-ordinates of a screw about which twists 
a rigid body whose co-ordinates are x , y , z , If, If, 1 3 2 , pf, pf, p 3 * with respect 
to the associated Cartesian axes, then the kinetic energy is Mu e 2 d 2 , where M 
is the mass, $ the twist velocity, and where 

V = a?e? + a?G? + b 2 3 2 + W + c 2 6> 5 2 + c 2 6 2 

+ 6^0 (^3 - #4) (0* + 0*) ~ ex. (0 5 - 6 ) (0 3 + t ) 
+ cy (6, - 6 ) (0, + 6,} - ay, (0, - 0,) (0, + 0.) 
+ az (0, - 2 ) (0 3 + 0.) - bz (0 3 - 4 ) (0, + 6> 2 ) 
+ i (/>! 2 - 2 ) (0 l + <9 2 ) 2 + i (pj - &) (0 t + 6tf + \ (p 3 2 - c 2 ) (0. + 6 ) 2 

- V (0* + 4 ) (O, + 0.) ~ ^ (0* + 0.) (0i + 0) - tt (0i + 0*) (0* + 04)- 

232 



356 THE THEORY OF SCREWS. [325- 

325. A Differential Equation satisfied by the Kinetic Energy. 

If the pitch p of the screw 6 about which a body is twisting receive a 
small increment 8p while the twist velocity is unaltered the change in 

kinetic energy is 

MpSpv*. 

But the addition of 8p to p has the effect ( 264) of changing each canonical 
co-ordinate 0^ into 



The variation thus arising in the kinetic energy equated to that already 
found gives the following differential equation which must be satisfied by uf, 

a (Of - 0f) + b (6? - 0f) + c (0? - Oft 



_ dul\ 03 + 04 fduf _ du 



G fduf _ duf\ 
\d0 5 d0J 



_ 
d0 1 d0 2b d0 3 d0 2c 

If we assume that u e - must be a rational homogeneous function of the 
second order in 1) ...0 6 we can, by solution of this equation, obtain the 
value of UQ- given in the last Article. 

326. Co-ordinates of Impulsive Screw in terms of the Instan 
taneous Screw. 

If a lt ... 6 be the canonical co-ordinates of an instantaneous screw and 
771, ... r) 6 the corresponding co-ordinates of the impulsive screw, then we have 

( ( J9)> 

1 du a * 1 duf , 

e?/! = - -j , e?; 2 = --- j , &c., 
a d^ a da 2 

and we obtain the following : 

+ ea^ = (+ pf + a 2 ) ^ (+ p? - a 2 ) a 2 + (az - bz - If) 3 

+ (az, + bz - 1 3 2 ) a 4 + (- ay + cy - I?) a 5 + (- ay, - cy - If) a c 

- eaT;, = (+ pi 2 - a 2 ) ^ (+ pf + a 2 ) 2 + (- az - bz - 1 3 Z ) a 3 

(- az + bz - I/) a 4 + (+ ay + cy - If) 6 + (+ ay, - cy - If) a G 
+ eby 3 = (+ az, - bz, - 1 3 -) a, + (- az - bz Q - If) a 2 + (+ pf + 6 2 ) a 3 

(+ p.? - b 2 ) 4 + (+ bx - cx - I*) a 5 + (6^0 + cx - /, 8 ) 6 

- ebr) 4 = (+ az Q -f bz - 1 3 ) a, + (- az + bz - If) 2 (+ p 2 2 - b 2 ) a 3 

(+ p./ -f b 2 ) 4 -f (- bx - cx - l^) a 5 + (- fo + cx - li 2 ) a ti 

+ ecvjr, = (- ay + cy - If) a a + (+ ay + cy - If) 2 + (+ bx - cx - If} <* 3 
(- bx - cx - If) a, + (+ pf + c 2 ) 5 (+ pf - c 2 ) 8 

- ecrj e = (- ay, - cy - If) a, + (ay, - cy - If) a 2 + (+ bx + cx - If) 3 

(- bx, + ex, - If) 4 + (+ pf - c 2 ) 5 + (pf + c 2 ) <V 



32 9 J VARIOUS EXERCISES. 357 

327. Another proof of article 303. 

As an illustration of the formulae just given we may verify a theorem of 
303 showing that when we know the instantaneous screw corresponding to 
a given impulsive screw, then a ray along which the centre of gravity must 
lie is determined. 

For subtracting the second equation from the first and repeating the 
process with each of the other pairs, we have 



- 2 ) + 2^ ( 3 + 4 ) - 2y (a-, + (i ), 
e (773 + *7 4 ) = 26 (ots - 04) + 2x ( 5 + a ) - 2^ (! + 02), 
e Ofc + %) = 2c (a e - ) + 2y (j + a 2 ) - 2# ( 3 + a 4 ). 

Eliminating e we have two linear equations in x^y^z^ thus proving the 
theorem. 

If we multiply these equations by a. l + a^, a 3 + a 4 , ar, + 6 respectively 
and add, we obtain 

e cos (OLVJ) = 2p a , 
thus giving a value for e. 

328. A more general Theorem. 

If an instantaneous screw be given while nothing further is known as to 
the rigid body except that the impulsive screw is parallel to a given plane 
A, then the locus of the centre of gravity is a determinate plane. 

Let A,, //,, v be the direction cosines of a normal to A, then 

At (^ 3 + ifc) + v (%, + i/) = 0, 



whence by substitution from the equations of the last Article we have a 
linear equation for X Q , y , z . 

329. Two Three-Systems. 

We give here another demonstration of the important theorem of .318, 
which states that when two arbitrary three-systems U and V are given, it is 
in general possible to design and place a rigid body in one way but only in 
one way, such that an impulsive wrench delivered on any screw 77 of V shall 
make the body commence to move by twisting about some screw a of U. 

Let the three principal screws of the system U have pitches a, b, c and 
take on the same three axes screws with the pitches a, b, c respectively. 
These six screws lying in pairs with equal and opposite pitches form the 
canonical co-reciprocals to be used. 



358 THE THEORY OF SCREWS. [329- 

As 77 belongs to the three-system V we must have the six co-ordinates of 
i) connected by three linear equations ( 77) ; solving these equations we have 

772 = ^77! +.8773 +775, 
774 = A i ll + B rj 3 + G rj s , 



The nine coefficients A, B, C, A , B , G , A", B", G" are essentially the co 
ordinates of the three-system V. We now seek the co-ordinates of the rigid 
body in terms of these quantities. 

Take the particular screw of U which has co-ordinates 

1, 0, 0, 0, 0, 0. 

Then the co-ordinates of the corresponding impulsive screw are 77!, 772, ... 
where 

+ ecu?! = p! 2 - a 2 ; + e&77 3 = az - bz - Z 3 2 ; + eC77 5 = - ay + cy - I? ; 
- ea77 2 = pj 2 - a 2 ; - &ii 4 = az + bz - / 3 2 ; - ec77 B = a y - cy Q - 1 2 2 . 

Since by hypothesis this is to belong to V, the following equations must 
be satisfied : 

2 ~ Pi 2 = A a *_P* + B az <> ~ bzp - 1 3 2 c cy - ay, - 1* 
a a b c 

- az - bz + 1 3 2 _ , ttM-p 2 az - bz - 1 3 2 cy - ay, - I? 

T -a- - h -O - j- ----- HO - - 

b c 



R ,, az - z - 



cy - ay - 



cab c 

In like manner by taking successively for a the screws wiih co-ordinates 

0, 0, 1, 0, 0, 
and 0, 0, 0, 0, 1, 0, 

we obtain six more equations of a similar kind. As these equations are 
linear they give but a single system of co-ordinates x , y 0> z , If, I*, 1 3 2 , 
P*> P*, Pa 2 for the rigid body. The theorem has thus been proved, for of course 
if three screws of U correspond to three screws of V then every screw in U 
must have its correspondent restricted to V. 

330. Construction of Homographic Correspondents. 

If the screws in a certain three -system U be the instantaneous screws 
whose respective impulsive screws form the three-system V, then when three 
pairs of correspondents are known the determination of every other pair of 
correspondents may be conveniently effected as follows. 



331] VARIOUS EXERCISES. 359 

We know ( 279) that to generate the unit twist velocity on an instan 
taneous screw a an impulsive wrench on the screw r\ is required, of which 
the intensity is 



COS 

the mass being for convenience taken as unity. 

Let a, ft, 7 be three of the instantaneous screws in U, and let 77, , be 
their respective impulsive screws in V. 

Let d, ft, 7 be the component twist velocities on a, ft, y of a twist velocity 
p on any other screw p, belonging to the system U. 

Then the impulsive wrench on V, which has p as its instantaneous screw 
will have as its components on 77, , f the respective quantities 

7 \ ^> ~ / 7T^\ P) "/ L,\ 7- 

cos (a?7) cos (p) cos (7^) 

These are accordingly the co-ordinates of the required impulsive wrench. 

331. Geometrical Solution of the same Problem. 

When three pairs of correspondents in the two impulsive and instan 
taneous systems of the third order V and U are known we can, in general, 
obtain the impulsive screw in V corresponding to any instantaneous screw p 
in U as follows. 

Choose any screw other than p in the three- system U and draw the 
cylindroid H through that screw arid p. Every screw on a cylindroid 
thus obtained must of course belong to U. Then H must have a screw in 
common with the cylindroid (a/3) drawn through a and ft, for this is 
necessarily true of any two cylindroids which lie in the same three-system. 
In like manner H must also have a screw in common with the cylindroid 
(ay) drawn through a and 7. But by the principle of 292 the several pairs 
of correspondents on the instantaneous cylindroid (aft) and the impulsive 
cylindroid (77 ) are determined. Hence the impulsive screw corresponding 
to one of the screws in H is known. In like manner the known pairs on the 
two cylindroids (ay) and (77^) will discover the impulsive screw corresponding 
to another instantaneous screw on H. As therefore we know the impulsive 
screws corresponding to two of the screws on H we know the cylindroid H 
which contains all the impulsive screws severally corresponding to instan 
taneous screws on H, of which of course p is one. But by 293 we can now 
correlate the pairs on H and H , and thus the required correspondent 
to p is obtained. 



360 THE THEORY OF SCREWS. [332, 

332. Co-reciprocal Correspondents in two Three-systems. 

If U be an instantaneous three-system and V the corresponding im 
pulsive three-system it is in general possible to select one set of three 
co-reciprocal screws in U whose correspondents in V are also co-reciprocal. 

As a preliminary to the formal demonstration we may note that the 
number of available constants is just so many as to suggest that some finite 
number of triads in U ought to fulfil the required condition. 

In the choice of a screw a in U we have, of course, two disposable 
quantities. In the choice of /3 which while belonging to U is further 
reciprocal to a there is only one quantity disposable. The screw belonging 
to U, which is reciprocal both to a and /?, must be unique. It is in fact 
reciprocal to five independent screws, i.e. to three of the screws of the system 
reciprocal to U, and to a and /3 in addition. 

We have thus, in general, neither more nor fewer than three disposable 
elements in the choice of a set of three co-reciprocal screws a, /3, 7 in U. 
This is just the number of disposables required for the adjustment of the 
three correspondents rj, , in V to a co-reciprocal system. We might, 
therefore, expect to have the number of solutions to our problem finite. We 
are now to show that this number is unity. 

Taking the six principal screws of inertia of the rigid body as the screws 
of reference, we have as the co-ordinates of any screw in U 



Xa 6 4- fA/3 6 -|- vy 6) 
where X, p,, v are numerical parameters. 

The co-ordinates of the corresponding screw in V are 

P! (Xctj + fji/Sj. + i/ 7l ), 
p 2 (Xa 2 + pfa 



where for symmetry p 1} . . . p 6 are written instead of + a, a, + b, b, &c. 
Three screws in U are specified by the parameters 

X , //, z/; X", //,", v"; X ", p", v". 
If these screws are reciprocal, we have 

= 2 P1 (Va, + /& + i/ 7l ) (X" ai + /*"& + *" 7l ), 
or = \ \"p a + n ^ pt + v v "p y + (X>" + XV) *r*p 

+ (XV + XV) OTay + (ii 
and two similar equations. 



333] VARIOUS EXERCISES. 361 

If the corresponding impulsive screws are reciprocal, then 

o = xv^v + /*y SfV/Si 2 + * v Spi v + (*>" + vy ) SiVaA 

+ (XV" + XV) SlvXTi + (pv" + ^ V 
and two similar equations. 

Take the two conies whose equations are 

= p a 

= a 



these conies will generally have a single common conjugate triangle. If the 
co-ordinates of the vertices of this triangle be X , ///, v ; X", // , z/ ; X ", //", v" \ 
then the equations just given in these quantities will be satisfied ; and as there 
is only one such triangle, the required theorem has been proved. 

It can be easily proved that a similar theorem holds good for a pair 
of impulsive and instantaneous cylindroids. 

333. Impulsive and Instantaneous Cylindroids. 

If a given cylindroid U be the locus of the screws about which 
a free rigid body would commence to twist if it had received an impulsive 
wrench about any screw on another given cylindroid V, it is required to 
calculate so far as practicable the co-ordinates of the rigid body. 

Let us take our canonical screws of reference so that the two principal 
screws of the cylindroid U have as co-ordinates 

1, 0, 0, 0, 0, 0, 
0, 0, 1, 0, 0, 0. 

The co-ordinates of any other screw on U will be 
a lt 0, 3 , 0, 0, 0. 

The cylindroid F will be determined by four linear equations in the 
co-ordinates of rj. These equations may with perfect generality be written 
thus ( 77), 



where A, B, A , B , A", B", A ", B " are equivalent to the eight co-ordinates 
defining the cylindroid V. 

The screw on U with co-ordinates 

1, 0, 0, 0, 0, 



362 THE THEORY OF SCREWS. [333, 

will have its corresponding impulsive screw defined by the equations 

( 326) 

+ ear]! = pi 2 + a? ; - ea^ = p x 2 - a 2 ; 

+ ebrjs = az bz - I? ; ebrj 4 = az + bz l s 2 ; 
+ ecrj 5 = cy - ay -1 2 2 ; - ecr) 6 = - ay - cy - Z 2 2 . 
By substituting these in the equations just given, we obtain 

az - bz - l s 2 _ . a? + pi* ^ a 2 - pi 2 
b a a 

- az n - bz + 1 3 2 , a? + p? , a 2 - Pl 2 



- - . - 
a 

cy - ay - I? = > , a 2 + pi 2 



ay + cy, + 1? _ A> ,, a? + p, 2 + B// , d? - pf 
c a a 

In like manner from the screw on U 

0, 0, 1, 0, 0, 
we obtain 

+ arj l = az Q bz ti I/ , ur)-> = az bz / 3 2 ; 

+ brj 3 = p, 2 + 6 2 ; -6174= p-?-b*; 

+ crj 5 = bx CX l*\ Crf = bx + CC If. 

Introducing these into the equations for 77, we have 

az - bz - I/ 7 az + bz + 1 3 2 
= A -- --- h x> 

a a 



6 2 - p 2 2 . , az - bz - 1 3 2 D , az + bz 
, r = A +n 



a a 

- bz - Z 3 2 az + bz 

r 



caa 
- bx - cx + /j 2 . , az - bz - l<? n ,,, az + bz + 1 3 * 

J\. i -*- 

c a a 

Thus we have eight equations while there are nine co-ordinates of the 
rigid body. This ambiguity was, however, to be expected because, as proved 
in 306, there is a singly infinite number of rigid bodies which stand to the 
two cylindroids in the desired relation. 

The equations, however, contain one short of the total number of co 
ordinates ; aa , y , z , I, 2 , I, 2 , 1 3 2 , p, 2 , p 2 2 are all present but p 3 2 is absent. 



334] VARIOUS EXERCISES. 363 

Hence from knowing the two cylindroids eight of the co-ordinates of 
the rigid body are uniquely fixed while the ninth remains quite indeter 
minate. Every value for p 3 2 will give one of the family of rigid bodies for 
which the desired condition is fulfilled. 

We have already deduced geometrically ( 306) the relations of these 
rigid bodies. We now obtain the same results otherwise. 

The momental ellipsoid around the centre of gravity has as its 
equation 

(x - x o y- pS +(y- 7/ ) 2 p.? + (z- ztf p s 2 - 2 (y - y ) (z - z,) I? 

- 2 (z - z ) (as - ) 4 2 - 2 (x - a? ) (y - y,} 1 3 2 - (yx, - xytf 

- ( z y - y z *f - ( xz - zx T = & 

This may be written in the form 

tf(z-ztf = R, 
where p 3 2 does not enter into R. 

As /> 3 2 varies this equation represents a family of quadrics which have 
contact along the section of R = by the plane z z^ = 0. This proves 
that a plane through the common centre of gravity and parallel to the 
principal plane of the cylindroid U passes through the conic along which 
the momental ellipsoids of all the different possible bodies have contact. All 
these quadrics touch a common cylinder along this conic. The infinite point 
on the axis of this cylinder is the pole of the plane z - Z Q for each quadric. 
Every chord parallel to the axis of the cylinder passes through this pole 
and is divided harmonically by the pole and the plane z z = 0. As the 
pole is at infinity it follows that in every quadric of the system a chord 
parallel to the axis of the cylinder is bisected by z z . Hence a diameter 
parallel to the axis of the cylinder is conjugate to the plane z z in every 
one of the quadrics. Thus by a different method we arrive at the theorems 
of 306. 

334. The Double Correspondents on Two Cylindroids. 

Referring to the remarkable homography between the impulsive screws 
on one cylindroid V and the corresponding instantaneous screws on another 
cylindroid U we have now another point to notice. 

If the screws on U were the impulsive screws, while those on V were the 
instantaneous screws, there would also have been a unique homography, the 
rigid bodies involved being generally distinct. 

But of course these homographies are in general quite different, that is to 
say, if A be a screw in U the instantaneous cylindroid, and B be its corre 
spondent in V the impulsive cylindroid, it will not in general be true that if 



364 THE THEORY OF SCREWS. [334- 

A be a screw in U the impulsive cylindroid, then B will be its instantaneous 
screw in V the instantaneous cylindroid. 

It is however to be now shown that there are two screws H 1 a,nd H 2 on U, 
and their correspondents K 1 and K 3 on V, which possess the remarkable 
characteristic that whether V be the impulsive cylindroid and U the instan 
taneous cylindroid or vice versa, in either case H l and K l are a pair of corre 
spondents, and so are H 2 and K z . 

Let B 1} B 2 , B 3 , &c. be the screws on U corresponding severally to the 
screws A l} A. 2 , A 3 , &c., on V when V is the impulsive cylindroid and U the 
instantaneous cylindroid. 

Let G l} C 2 , C 3 , &c., be the screws on U corresponding severally to the 
screws A l} A 2 , A 3 , &c., on V when U is now the impulsive cylindroid, and V 
the instantaneous cylindroid. 

The systems A lt A 3 , A,, &c., and B,, B.,, B 3 , &c., are homographic. 
The systems C lt a , C s , &c., and A lt A s , A 3 , &c., are homographic. 

Hence also, 

The systems B lt B. 2 , B 3 , &c., and C lt C*, C 3 , &c. are homographic. 

Let 7/j, HZ be the two double screws on U belonging to this last homo- 
graphy, then their correspondents K l} K 2 on V will be the same whether U 
be the impulsive cylindroid and V the instantaneous cylindroid or vice versa. 

There can be no other pairs of screws on the two cylindroids possessing 
the same property. 

335. A Property of Co-reciprocals. 

Let a, /3, 7 be any three co-reciprocal screws. If 77, ff, are the three 
screws on which impulsive wrenches would cause a free rigid body to twist 
about a, /3, 7 respectively, then 

cos (af ) cos (17) cos ( 7 ) + cos (a) cos () cos (77;) = 0. 
We have from 281 the general formula 



cos fy + cos 



cos a?; cos 



but as a and /3 are reciprocal each side of this equation must be zero. We 
thus have 

Pa 



cos , = - -- cos 



^ -/ 

cos (017) cos 



337] VARIOUS EXERCISES. 365 

and similarly, 



COS (yf;) = --,- _ COS 

COS ( 7 ) 

Pv i v\ P- 

/ _ cos(a) = . N cos 

cos ( 7 ) cos (a?;) 

whence we obtain 



cos (a) cos (fir)) cos ( 7 f ) + cos (a) cos (/3) cos (yrf) = 0, 

for it is shown in 283 that p a -:- cos (a??) or the other similar expressions can 
never be zero. 

336. Instantaneous Screw of Zero Pitch. 

Let a be an instantaneous screw of zero pitch. Let two of the canonical 
co-reciprocals lie on a, then the co-ordinates of a are 

i, i o, o, o, o. 

The co-ordinates of the impulsive screw ij are given by the formulae of 
32G which show that 



We thus have 

(i + Oa) (Vi + fc) + ( 3 + 04) (ife + ^4) + ( 5 + ) (77, + 77 6 ) = 0, 

which proves what we already knew, namely, that a and 77 are at right angles 
(S 293). 

We also have 

2/0 (l?3 + ^4) + ^0 (l?5 + %) = 0, 

which proves the following theorem : 

If the instantaneous screw have zero-pitch then the centre of gravity of 
the body lies in the plane through the instantaneous screw and perpendicular 
to the impulsive screw. 

337. Calculation of a Pitch Quadric. 

If a, fi, 7 be three instantaneous screws it is required to find with respect 
to the principal axes through the centre of gravity, the equation to the pitch 
quadric of the three-system which contains the three impulsive screws corre 
sponding respectively to a, @, 7 . The co-ordinates of these screws are 
expressed with reference to the six principal screws of inertia. 

We make the following abbreviations : 

A = a? (a* - 2 2 ) + b* ( 3 2 - 4 2 ) + c 3 ( 5 2 - a 6 2 ), 
B = a? (&*- /3 2 2 ) + 6 3 (/3 3 - - &*) + c s (& 2 - & a ), 
C=a? ( 7l 2 - 



366 THE THEORY OF SCREWS. [337 

P = + bcx ((A - A) (73 - 74) ~ (A - A) ( 7s - 7.)), 

+ ca y ((75 7) (& ~ &) ~~ (71 ~ 72) (& A))> 

Q = + bcx ((a, - a 4 ) (75 - 7 6 ) - (a 5 - a 6 ) (73 - 7,)), 
+ acy ((71 - 7 2 ) ( 5 - 6 ) - (i - <* 2 ) (75 - 7e)) 
+ abz ((! - 2 ) (73 - 74) - (a, - 4 ) (7! - 7,,)) ; 

- a a) (A" A) (& /3 2 ) ( 5 tte)), 

^ = a 3 (: + a) (A - &) + 6 s (a, + a 4 ) (A - A) + c 3 (* 5 + a 6 ) (/3 6 - /3 6 ), 
Z^ a = a 3 (A + /3 2 ) (aj - 2 ) + 6 3 (/3 3 + A) ( 8 - "4) + c 3 (A + A) ( a s - e) I 
Z ay = a 3 (d + a) (71 - 72) + b 3 ( 3 + 4 ) (73 - 74) + c 3 (cc 5 + 6 ) (75 - 7), 

^ya = tt 3 (7x + 7 2 ) (! - a s ) + 6 3 (7s + 74) (3 ~ 4 ) + C 3 (75 + 7e) (s - a e) I 

X 0y = a 3 (A + A) (7i - 72) + ^ 3 (A + &) (73 - 74) + c 3 (A + A) (7s ~ 7e), 



The required equation is then as follows : 



= 



A, -R + L^, 
R + L afi , B, - 

Q + L ay , P + Lfiy , 



C 



CHAPTER XXIV. 

THE THEORY OF SCREW-CHAINS*. 

338. Introduction. 

In the previous investigations of this volume the Theory of Screws has 
been applied to certain problems in the dynamics of one rigid body. I 
propose to show in the present chapter to what extent the conceptions 
and methods of the Theory of Screws may be employed to elucidate certain 
problems in the dynamics of any material system whatever. 

By such a system I mean an arbitrary arrangement of /j, rigid bodies 
of any form or construction, each body being either entirely free or con 
strained in any manner by relations to fixed obstacles or by connexions 
of any kind with one or more of the remaining p 1 pieces. 

For convenience we may refer to the various bodies in the system by 
the respective numerals 1, 2, ... /n. This numbering may be quite arbitrary, 
and need imply no reference whatever to the mechanical connexions of 
the pieces. The entire set of material parts I call for brevity a mass-chain, 
and the number of the bodies in a mass-chain may be anything from unity 
to infinity. 

I write, as before, of only small movements, but even with this limitation 
problems of equilibrium, of small oscillations and of impulsive movements 
are included. By the order of the freedom of the mass-chain, I mean the 
number of generalized co-ordinates which would be required to specify a 
position which that mass-chain was capable of assuming. The order cannot 
be less than one (if the mass- chain be not absolutely fixed), while if each 
element of the mass-chain be absolutely free, the order will be as much 
as 6/i. 

Starting from any arbitrary position of the mass-chain, let it receive a 
small displacement. Each element will be displaced from its original 
position to an adjacent position, compatible of course with the conditions 

* Transactions Royal Irish Acad. Vol. xxvm. p. 99 (1881). 



368 THE THEORY OF SCREWS. [338- 

imposed by the constraints. The displacement of each element could, 
however, have been effected by a twist of appropriate amplitude about a 
screw specially correlated to that element. The total effect of the displace 
ment could, therefore, have been produced by giving each element a certain 
twist about a certain screw. 

339. The Graphic and Metric Elements. 

In the lowest type of freedom which the mass-chain can possess (short 
of absolute fixity) the freedom is of the first order, and any position of 
the mass-chain admits of specification by a single co-ordinate. In such a 
case the screw appropriate to each element is unique, and is completely 
determined by the constraints both in position and in pitch. The ratio 
of the amplitude of each twist to the amplitudes of all the other twists is 
also prescribed by the constraints. The one co-ordinate which is arbitrary 
may be conveniently taken to be the amplitude of the twist about the 
first screw. To each value of this co-ordinate will correspond a possible 
position of the mass-chain. As the ratios of the amplitudes are all known, 
and as the first amplitude is given, then all the other amplitudes are known, 
and consequently the position assumed by every element of the mass-chain 
is known. 

The whole series of screws and the ratios of the amplitudes thus embody 
a complete description of the particular route along which the mass-chain 
admits of displacement. The actual position of the mass-chain is found 
by adding to the purely graphic element which describes the route a metric 
element, to indicate the amplitude through which the mass-chain has 
travelled along that route. This amplitude is the arbitrary co-ordinate. 

340. The Intermediate Screw. 

It will greatly facilitate our further progress to introduce a conventional 
process, which will clearly exhibit the determinate character of the ratios 
of the amplitudes in the screw series. Consider the two first screws, i and or 2 
of the series. Draw the cylindroid (o. l , 2 ) which contains these two screws. 
Since a l and 2 are appropriated to two different elements of the mass-chain, 
no kinematical significance can be attached to the composition of the two 
twists on j and 2 . If, however, the two twists on j and a 2 , having the 
proper ratio of amplitudes, had been applied to a single rigid body, the dis 
placement produced is one which could have been effected by a single 
twist about a single screw on the cylindroid (a l} et 2 ). If this inter 
mediate screw be given, the ratio of the amplitudes of the twists on 
the given screws is determined. It is in fact equal to the ratio of the 
sines of the angles into which the intermediate screw divides the angle 
between the two given screws. With a similar significance we may conceive an 
intermediate screw inserted between every consecutive pair of the p original 
screws. 



342] THE THEORY OF SCREW-CHAINS. 369 

341. The definition of a Screw-chain. 

It will be convenient to have a name which shall concisely express the 
entire series of /A original screws with the //, 1 intermediate screws whose 
function in determining the amplitudes has just been explained. We may 
call it a screw-chain. A twist about a screw-chain will denote a displace 
ment of a mass-chain, produced by twisting each element about the 
corresponding screw, through an amplitude whose ratio to the amplitudes 
on the two adjacent screws is indicated by the intermediate screws. The 
amplitude of the entire twist will, as already mentioned, be most conveniently 
expressed by the twist about the first screw of the chain. We hence have 
the following statement : 

The most general displacement of which a mass-chain is capable can be 
produced by a twist about a screw-chain. 

342. Freedom of the first order. 

Given a material system of p elements more or less connected inter se, 
or related to fixed points or supports : let it be required to ascertain the 
freedom which this material system or mass-chain enjoys. The freedom is to be 
tested by the capacity for displacement which the mass-chain possesses. As 
each such displacement is a twist about a screw-chain, a complete examina 
tion of the freedom of the mass-chain will require that a trial be made to 
twist the mass-chain about every screw-chain in space which contains the 
right number of elements /u,. If in the course of these trials it be found that 
the mass-chain cannot be twisted about any screw-chain, then the system 
is absolutely rigid, and has no freedom whatever. If after all trials have 
been made, one screw-chain, and only one, has been discovered, then the 
mass-chain has freedom of the first order, and we have the result thus 
stated : 

When a mass-chain is so limited by constraints, that its position can be 
expressed by a single co-ordinate, then the mass-chain is said to have freedom 
of the first order, and its possible movements are solely those which could be 
accomplished by twisting about one definite screw-chain. 

By this method of viewing the question we secure the advantage of 
eliminating, as it were, the special characters of the constraints. The 
essential moving parts of a steam-engine, for example, have but one degree 
of freedom. Each angular position of the fly-wheel necessarily involves a 
definite position of all the other parts. A small angular motion of the fly 
wheel necessarily involves a definite small displacement of each of the 
other parts. Complicated as the mechanism may be, it is yet always possible 
to construct a screw-chain, a twist about which would carry each element 
from its original position into the position it assumes after the displacement 
has been effected. 

B. 24 



370 THE THEORY OF SCREWS. [343 

343. Freedom of the second order. 

Suppose that after one screw-chain has been discovered, about which the 
mass-chain can be twisted, the search is continued until another screw-chain 
is detected of which the same can be asserted. We are now able to show, 
without any further trials whatever, that there must be an infinite number 
of other screw -chains similarly circumstanced. For, compound a twist of 
amplitude a on one chain, a, with the twist of amplitude /3 on the other, /?. 
The position thus attained could have been attained by a twist about some 
single chain 7. As a and /3 are arbitrary, it is plain that 7 can be only one 
of a system of screw-chains at least singly infinite in number about which 
twisting must be possible. 

The problem to be considered may be enunciated in a somewhat more 
symmetrical manner, as follows : 

To determine the relations of three screw-chains, a, /3, 7, such that if 
a mass-chain be twisted with amplitudes of, ft , 7 , about each of these screw- 
chains in succession, the mass-chain will regain the same position after the 
last twist which it had before the first. 

This problem can be solved by the aid of principles already laid down 
(Chap. H.). Each element of the mass-chain receives two twists about 
a and /3 ; these two twists can be compounded into a single twist about 
a screw lying on the cylindroid defined by the two original screws. We 
thus have for each element a third screw and amplitude by which the required 
screw-chain 7 and its amplitude 7 can be completely determined. 

A mass-chain free to twist to and fro on the chains a and /3 will therefore 
be free to twist to and fro on the chain 7. These three chains being known, 
we can now construct an infinite number of other screw-chains about which 
the mass-chain must be also able to twist. 

Let 8 be a further screw-chain of the system, then the screws OL I} /3 1( 7^ 8 1 
which are the four first screws of the four screw-chains must be co- 
cylindroidal ; so must 2 > A, y 2 , 8 2 and each similar set. We thus have ^ 
cylindroids determined by the two first chains, and each screw of every chain 
derived from this original pair will lie upon the corresponding cylindroid. 
We have explained ( 125) that by the anharmonic ratio of four screws on 
a cylindroid we mean the anharmonic ratio of a pencil of four lines parallel 
to these screws. If we denote the anharmonic ratio of four screws such as 
i> &, 7i> Si b y tne symbol 

[i, fii, 71, $1], 

then the first theorem to be now demonstrated is that 

[i, &, 71, S,]=[ 2 , &, 72> &>] = &c. = [a M , $1, 7 M , M ], 
or that the anharmonic ratio of each group is the same. 



343] THE THEORY OF SCREW-CHATNS. 371 

This important proposition can be easily demonstrated by the aid of 
fundamental principles. 

The two first chains, a and /3, will be sufficient to determine the entire 
series of cylindroids. When the third chain, 7, is also given, the construction 
of additional chains can proceed by the anharmonic equality without any 
further reference to the ratios of the amplitudes. 

When any screw, 8 l} is chosen arbitrarily on the first cylindroid, then 
8 2 , 6 3 , &c., ... 8^, are all determined uniquely; for a twist about ^ can be 
decomposed into twists about x and &. The amplitudes of the twists on 
ofj and & determine the amplitudes on or 2 and /3 2 by the property of the 
intermediate screws which go to make up the screw-chains, and by com 
pounding the twists on a., and /8 2 we obtain S 2 . If any other screw of the 
series, for example, 8 2 , had been given, then it is easy to see that ^ and 
all the rest, 8 3 , ... B^, are likewise determined. Thus for the two first 
cylindroids, we see that to any one screw on either corresponds one screw 
on the other. 

If one screw moves over the first cylindroid then its correspondent will 
move over the second and it will now be shown that these two screws trace 
out two homographic systems. Let us suppose that each screw is specified 
by the tangent of the angle which it makes with one of the principal screws 
of its cylindroid. Let l , fa be the angles for two corresponding screws 
on the first arid second cylindroids, then we must have some relation which 
connects tan 1 and tan^j. But this relation is to be consistent with the 
condition that in every case one value of tan l is to correspond to one 
value of tan fa, and one value of tan fa to one value of tan lf 

If for brevity we denote tan 6 l by x and tan fa by x then the geometrical 
conditions of the system will give a certain relation between x and x . The 
one-to-one condition requires that this relation must be capable of being 
expressed in either of the forms 

x=U ; x =U, 

where U is some function of x and where U is a function of x. From the 
nature of the problem it is easily seen that these functions are algebraical 
and as they must be one valued they must be rational. If we solve the first 
of these equations for x the result that we obtain cannot be different from 
the second equation. The first equation must therefore contain x only in the 
first degree in the form (see Appendix, Note 7) 



/ / / 

px +q 

The relation between tan 0^ and tan fa will therefore have the form which 
may generally be thus expressed, 

a tan 1 tan fa+b tan l + c tan <f> 1 + d = 0. 

242 



372 THE THEORY OF SCREWS. [343 

Let &!, 6.,, :j , 4 be the angles of four screws on the first cylindroid, then the 
anharmonic ratio will be 



sin (0 4 -6> 3 ) sin (02-00 

From the relation just given between tan0 1( tau^, which applies of course 
to the other corresponding pair, it will be easily seen that this anharmonic 
ratio is unaltered when the angles < 1; < 2 , &c., are substituted for 1? 2 , &c. 

We have, therefore, shown that the anharmonic ratio of four screws on 
the first cylindroid is equal to that of the four corresponding screws on the 
second cylindroid, and so on to the last of the //, cylindroids. 

As soon, therefore, as any arbitrary screw Sj has been chosen on the first 
cylindroid, we can step from one cylindroid to the next, merely guided in 
choosing S 2 , S 3 , &c., by giving a constant value to the anharmonic ratio of 
the screw chosen and the three other collateral screws on the same cylindroid. 
Any number of screw-chains belonging to the system may be thus readily 
constructed. 

This process, however, does not indicate the amplitudes of the twists 
appropriate to S lt S 2 , & 3 , &c. One of these amplitudes may no doubt be 
chosen arbitrarily, but the rest must be all then determined from the 
geometrical relations. We proceed to show how the relative values of these 
amplitudes may be clearly exhibited. 

The first theorem to be proved is that in the three screw-chains a, ft, 7 
the screws intermediate to ! and 2 , to /3j and (3. 2 , to y 1 and j 2 are co- 
cylindroidal. This important step in the theory of screw-chains can be 
easily inferred from the fundamental property that three twists can be 
given on the screw-chains a, /3, 7, which neutralize, and that consequently 
the three twists on the screws a 1 , j3 1} <y l will neutralize, as will also those 
on a 2 , /3 2 , 72- These six twists must neutralize when compounded in any 
way whatever. We shall accordingly compound a x and a 2 into one twist 
on their intermediate screw, and similarly for /^ and /3 2 , and for y l and 73. 
We hence see that the three twists about the three intermediate screws 
must neutralize, and consequently the three intermediate screws must be 
co-cylindroidal. 

We thus learn that in addition to the several cylindroids containing the 
primary screws of each of the system of screw-chains about which a mass- 
chain with two degrees of freedom can twist, there are also a series of 
secondary cylindroids, on which will lie the several intermediate screws of 
the system of screw-chains. 

If Sj be given, then it is plain that the intermediate screw between Sj 
and 8 2 , as well as all the other screws of the chain and their intermediate 



343] THE THEORY OF SCREW-CHAINS. 373 

screws, can be uniquely determined. If, however, the intermediate screw 
between S x and 8 2 be given, the entire chain 8 is also determined, yei it is 
not immediately obvious that that determination is unique. We can, however, 
show as follows that this is generally the case. 

Let S 12 denote the given intermediate screw, and let this belong, not 
only to the chain S 1} 8 2 , &c., but to another chain &/, 8 2 , &c. We then 
have S 1} <5 12 , 8. 2 co-cylindroidal, and also S/, 8^, 8 2 co-cylindroidal. Decom 
pose any arbitrary twist of amplitude 6 on S 12 into twists on 8 L and 2 , and a 
twist of amplitude 6 on the same 8 12 into twists on 8/ and S 2 . Then the 
four twists must neutralize ; but the two twists on S/ and S t compound into 
a twist on a screw on the first cylindroid of the system ; and / and 6 2 into 
a twist on the second cylindroid of the system ; and as these two resultant 
twists must be equal and opposite it follows that they must be on the same 
screw, and that, therefore, the cylindroids belonging to the first and second 
elements of the system must have a common screw. It is, however, not 
generally the case that two cylindroids have a common screw. It is only true 
when the two cylindroids are themselves included in a three-system, this 
could only arise under special circumstances, which need not be further con 
sidered in a discussion of the general theory. 

It follows from the unique nature of the correspondence between the 
intermediate cylindroids and the primary cylindroids that one screw on any 
cylindroid corresponds uniquely to one screw on each of the other cylindroids; 
the correspondence is, therefore, homographic. 

We have now obtained a picture of the freedom of the second order of the 
most general type both as to the material arrangement and the character of 
the constraints : stating summarily the results at which we have arrived, 
they are as follows : 

A mass-chain of any kind whatever receives a small displacement. This 
displacement is under all circumstances a twist about a screw-chain. If 
the mass-chain admits of a displacement by a twist about a second screw- 
chain, then twists about an infinite number of other screw-chains must also 
be possible. To find, in the first place, a third screw-chain, give the mass- 
chain a small twist about the first chain ; this is to be followed by a small 
twist about the second chain : the position of the mass-chain thus attained 
could have been reached by a twist about a third screw-chain. The system 
must, therefore, be capable of twisting about this third screw-chain. When 
three of the chains have been constructed, the process of finding the re 
mainder is greatly simplified. Each element of the mass-chain is, in each 
of the three displacements just referred to, twisted about a screw. These 
three screws lie on one cylindroid appropriate to the element, and there are 
just so many of these cylindroids as there are elements in the mass-chain. 



374 THE THEORY OF SCREWS. [343- 

Betvveen each two screws of a chain lies an intermediate screw, introduced 
for the purpose of defining the ratio of the amplitudes of the two screws of 
the chain on each side of it. In the three chains two consecutive elements 
will thus have three intermediate screws. These screws are co-cylindroidal. 
We thus have two series of cylindroids : the first of these is equal in number 
to the elements of the mass-chain (/it), each cylindroid corresponding to one 
element. The second series of cylindroids consists of one less than the 
entire number of elements (^ - 1). Each of these latter cylindroids corre 
sponds to the intermediate screw between two consecutive elements. An 
entire screw-chain will consist of fju primary screws, and //, 1 intermediate 
screws. To form such a screw-chain it is only necessary to inscribe on each 
of the 2yu, 1 cylindroids a screw which, with the other three screws on that 
cylindroid, shall have a constant anharmonic ratio. Any one screw on any 
one of the 2/^1 cylindroids may be chosen arbitrarily ; but then all the 
other screws of that chain are absolutely determined, as the anharmonic 
ratio is known. The mass-chain which is capable of twisting about two 
screw-chains cannot refuse to be twisted about any other screw-chain con 
structed in the manner just described. It may, however, refuse to be 
twisted about any screw-chains not so constructed ; and if so, then the 
mass-chain has freedom of the second order. 

344. Homography of Screw-systems. 

Before extending the conception of screw-chains to the examination of 
the higher orders of freedom, it will be necessary to notice some extensions 
of the notions of homography to the higher orders of screw systems. On 
the cylindroid the matter is quite simple. As we have already had occasion 
to explain, we can conceive the screws on two cylindroids to be homo- 
graphically related, just as easily as we can conceive the rays of two plane 
pencils. The same ideas can, however, be adapted to the higher systems 
of screws the 3rd, the 4th, the 5th while a case of remarkable interest 
is presented in the homography of two systems of the 6th order. 

The homography of two three-systems is completely established when to 
each screw on one system corresponds one screw on the other system, and 
conversely. We can represent the screws in a three-system by the points 
in a plane (see Chap. xv.). We therefore choose two planes, one for each 
of the three-systems, and the screw correspondence of which we are in 
search is identical with the homographic point-correspondence between the 
two planes. 

We have already had to make use in 317 of the fundamental property 
that when four pairs of correspondents in the two planes are given then 
the correspondence between every other pair of points is determined by 



345] THE THEORY OF SCREW-CHAINS. 375 

rigorous construction. Any fifth point in one plane being indicated, the 
fifth point corresponding thereto in the other plane can be determined. 
It therefore follows that when four given screws on one three-system are 
the correspondents of four indicated screws on the other system, then the 
correspondence is completely established, and any fifth screw on one system 
being given, its correspondent on the other is determined. 

345. Freedom of the third order. 

We are now enabled to study the small movements of any mass-chain 
which has freedom of the third order. Let such a mass-chain receive 
any three displacements by twists about three screw-chains. It will, of 
course, be understood that these three screw-chains are not connected in 
the specialized manner we have previously discussed in freedom of the 
second order. In such a case the freedom of the mass-chain would be of 
the second order only and not of the third. The three screw-chains now 
under consideration are perfectly arbitrary ; they may differ in every con 
ceivable way, all that can be affirmed with regard to them is that the 
number of primary screws in each chain must of course be equal to fj,, 
i.e. to the number of material elements of which the mass-chain consists. 

It may be convenient to speak of the screws in the different chains which 
relate to the same element (or in the case of the intermediate screws, the 
same pair of elements) as homologous screws. Each set of three homologous 
screws will define a three-system. Compounding together any three twists 
on the screw-chains, we have a resultant displacement which could have 
been effected by a single twist about a fourth screw-chain. The first theorem 
to be proved is, that each screw in this fourth screw-chain must belong to the 
three-system which is defined by its three homologous screws. 

So far as the primary screws are concerned this is immediately seen. 
Each element having been displaced by three twists about three screws, the 
resultant twist must belong to the same three-system, this being the im 
mediate consequence of the definition of such a system. Nor do the inter 
mediate screws present much difficulty. It must be possible for appropriate 
twists on the four screw-chains to neutralize. The four twists which the 
first element receives must neutralize : so must also the four twists imparted 
to the second element. These eight twists must therefore neutralize, 
however they may be compounded. Taking each chain separately, these 
eight twists will reduce to four twists about the four intermediate screws : 
these four twists must neutralize ; but this is only possible if the four 
intermediate screws belong to a three-system. 

On each of fj, primary three-systems, and on each of yu, 1 intermediate 
three-systems four screws are now supposed to be inscribed. We are to 



376 THE THEORY OF SCREWS. [345, 

determine a fifth screw about which the system even though it has only 
freedom of the third order, must still be permitted to twist. 

To begin with we may choose an arbitrary screw in any one of the three- 
systems. In the exercise of this choice we have two degrees of latitude; 
but once the choice has been made, the remainder of the screw-chain is 
fixed by the following theorem : 

If each set of five homologous screws of five screw-chains lies on a three- 
system, and if a mass-chain be free to twist about four of these screw-chains, 
it will also be free to twist about the fifth, provided each set of homologous 
screws is homographic with every other set. 

Let 8 denote the fifth screw-chain. If 8 l be chosen arbitrarily on the 
three-system which included the first element, then a twist about 8 l can 
be decomposed into three twists on a t> /8 lf 7,. By the intermediate screws 
these three twists will give the amplitudes of the twists on all the other 
screws of the chains a, /3, y, and each group of three homologous twists 
being compounded, will give the corresponding screws on the chain 8. We 
thus see that when 8, is given, 8 2 , 8 3 , &c., are all determinate. It is also 
obvious that if S 2 , or any other primary screw of the chain, were given, then 
all the other screws of the chain would be determined uniquely. 

If, however, an intermediate screw, 8 12 , had been given, then, although 
the conditions are, so far as number goes, adequate to the determination of 
the screw-chain, it will be necessary to prove that the determination is 
unique. This is proved in the same manner as for freedom of the second 
order ( 343). If there were two screw-chains which had the same inter 
mediate screw, then it must follow that the two primary three-systems must 
have a common screw, which is not generally the case. 

We have thus shown that when any one screw of the chain 8, whether 
primary or intermediate, is given, then all the rest of the screws of the 
chain are uniquely determinate. Each group of five homologous screws must 
therefore be homographic. 

It is thus easy to construct as many screw-chains as may be desired, 
about which a mass-chain which has freedom of the third order must be 
capable of twisting. It is only necessary, after four chains have been 
found, to inscribe an arbitrary screw on one of the three-systems, and then 
to construct the corresponding screw on each of the other homologous 
systems. 

In choosing one screw of the chain we have two degrees of latitude: we 
may, for example, move the screw chosen over the surface of any cylindroid 
embraced in the three-system: the remaining screws of the screw-chain, 
primary and intermediate, will each and all move over the surface of corre 
sponding cylindroids. 



346] THE THEORY OF SCREW-CHAINS. 377 

If the mass-chain cannot be twisted about any screw- chain except those 
we have now been considering, then the mass-chain is said to have freedom 
of the third order. If, however, a fourth screw-chain can be found, about 
which the system can twist, and if that screw-chain does not belong to the 
doubly infinite system just described, then the mass-chain must have freedom 
of at least the fourth order. 



346. Freedom of the fourth order. 

The homologous screws in the four screw-chains about which the mass- 
chain can twist form each a four-system. All the other chains which can 
belong to the system m.ust consist of screws, one of which lies on each of the 
four-systems. 

It will facilitate the study of the homography of two four-systems to 
make use of the analogy between the homography of two spaces and the 
homography of two four-systems as already we had occasion to do in 317. 
A screw in a four-system is defined by four homogeneous co-ordinates 
whereof only the ratios are significant. Each screw of such a system can 
therefore be represented by one point in space. The homography of two 
spaces will be completely determined if five points, a, b, c, d, e in one space, 
and the five corresponding points in the other space, a, b , c , d , e are 
given. 

From the four original screw-chains we can construct a fifth by com 
pounding any arbitrary twists about two or more of the given chains. When 
five chains have been determined, then, by the aid of the principle of homo 
graphy, we can construct any number. 

That each set of six homologous screws is homographic with every other 
set can be proved, as in the other systems already discussed. With respect 
to the intermediate screws a different proof is, however, needed to show 
that when one of these screws is given the rest of the chain is uniquely de 
termined. The proof we now give is perhaps simpler than that previously 
used, while it has the advantage of applying to the other cases as well. Let 
a, /3, 7, 8 be four screw-chains, and let e I2 , an intermediate screw of the 
chain e, be given. We can decompose a twist on e 12 into components of 
definite amplitude on a I2 , &,, 7,3, 8 K . The first of these can be decomposed 
into twists on c^ and a. 2 ; the second on & and /3 2 , &c. Finally, the four 
twists on !, &, 7!, 8 1 can be compounded into one twist, e,, and those on 
2, &, 7 2 , 8-2 compounded into a twist on e 2 . In this way it is obvious that 
when e 12 is given, then ej and e 2 are uniquely determined, and of course the 
same reasoning applies to the whole of the chain. We thus see that when 
any screw of the chain is known, then all the rest are uniquely determined, 
and therefore the principle of homography is applicable. 



378 THE THEORY OF SCREWS. [346- 

In the choice of a screw-chain about which a mass-chain with four 
degrees of freedom can twist there are three arbitrary elements. We may 
choose as the first screw of the chain any screw from a given four-system. 

If one screw of the chain be moved over a two-system, or a three-system 
included in the given four-system, then every other screw of the chain will 
also describe a corresponding two-system or three-system. 

347. Freedom of the fifth order. 

In discussing the movements of a system which has freedom of the fifth 
order, the analogies which have hitherto guided us appear to fail. Homo- 
graphic pencils, planes, and spaces have exhibited graphically the relations 
of the lower degrees of freedom ; but for freedom of the fifth degree these 
illustrations are inadequate. No real difficulty can, however, attend the 
extension of the principles we have been considering to the freedom of the 
fifth order. We can conceive that two five-systems are homographically 
related, such that to each screw on the one corresponds one screw on the 
other, and conversely. To establish the homography of the two systems it 
will be necessary to know the six screws on one system which correspond to 
six given screws on the other : the screw in either system corresponding to 
any seventh screw in the other is then completely determined. 

In place of the methods peculiar to the lower degrees of freedom, we 
shall here state the general analytical process which is of course available in 
the lower degrees of freedom as well. 

A screw 6 in a five-system is to be specified by five co-ordinates 1 , # 2 , B 3 , 
64, B 5 . These co-ordinates are homogeneous; but their ratios only are con 
cerned, so they are equivalent to four data. The five screws of reference 
may be any five screws of the system. Let <f> be the screw of the second 
system which is to correspond to 6 in the first system. The co-ordinates of 
<f> may be referred to any five screws chosen in the second system. It will 
thus be seen that the five screws of reference for <j> are quite different from 
those of 6. 

The geometrical conditions expressing the connection between </> and 9 
will give certain equations of the type 

where t/j, ..., U 5 are homogeneous functions of B lt ..., 9 5 . These equations 
express that one 9 determines one <f>. As however one < is to determine one 
9 we must have also equations of the type 

where //, ..., U 6 are functions of <f> 1} ..., </> 5 . 

From the nature of the problem these functions are algebraical and as 
they must be one valued they must be rational functions. We have therefore 



348] THE THEORY OF SCREW-CHAINS. 379 

a case of "Rational Transformation" (see Salmon s Higher Plane Curves, 
Chap. VIII.). The theory is however here much simplified. In this case 
none of the special solutions are admissible which produce the critical cases. 
Consider the equations U. 2 = Q, ..., U 5 = 0. They will give a number of 
systems of values for lf ..., 9 5 equal to the product of the degrees of 
C/2 > U 5 . Each of these 6 screws would be a correspondent to the same 
<j) screw 1, 0, 0, 0, 0. But in the problems before us this <f> as every other < 
can have only one correspondent. Hence all the functions U lt U 2 , etc. 
Ui, 17%, etc. must be linear. We may express the first set of equations thus : 

fa = (11) 0, + (12) a + (13) e 3 + (14) 4 + (15) 6 t , 
fa = (21) 1 + (22) 2 + (23) e. A + (24) 0, + (25) 9 5 , 
fa = (31) 0, + (32) 9, + (33) 3 + (34) 6, + (35) t , 
fa = (41) 0, + (42) 0, + (43) 3 + (44) 4 + (45) t , 
fa = (51) 0, + (52) 6, + (53) B, + (54) 9, + (55) . 

For the screw (j> to be known whenever 9 is given, it will be necessary to 
determine the various coefficients (11), (12), &c. These are to be determined 
from a sufficient number of given pairs of corresponding screws. Of these co 
efficients there are in all twenty-five. If we substitute the co-ordinates of one 
given screw 9, we have five linear equations between the co-ordinates. Of 
these equations, however, we can only take the ratios, for each of the co 
ordinates may be affected by an arbitrary factor. Each of the given pairs 
of screws will thus provide four equations to aid in determining the co 
efficients. Six pairs of screws being given, we have twenty-four equations 
between the twenty-five coefficients. These will be sufficient to determine 
the ratios of the coefficients. We thus see that by six pairs of screws the 
homography of two five-systems is to be completely defined. To any seventh 
screw on one system corresponds a seventh screw on the other system, which 
can be constructed accordingly. 

348. Application of Parallel Projections. 

It will, however, be desirable at this point to introduce a somewhat 
different procedure. We can present the subject of homography from 
another point of view, which is specially appropriate for the present theory. 
The notions now to be discussed might have been introduced at the outset. 
It was, however, thought advantageous to concentrate all the light that 
could be obtained on the subject ; we therefore used the point-homography 
of lines, of planes, and of spaces, so long as they were applicable. 

The method which we shallnow adopt is founded on an extension of 
what are known as " parallel projections " in Statics. We may here recall 



380 THE THEORY OF SCREWS. [348 

the outlines of this theory, with the view of generalizing it into one adequate 
for our purpose. 

We can easily conceive of two systems of corresponding forces in two 
planes. To each force in one plane will correspond one force in the other 
plane, and vice versa. To any system of forces in one plane will correspond 
a system of forces in the other plane. We are also to add the condition that 
if one force x vanishes, the corresponding force x will also vanish. 

The fundamental theorem which renders this correspondence of im 
portance is thus stated : 

If a group offerees in one of the planes would equilibrate when applied 
to a rigid body, then the corresponding group of forces in the other plane 
would also equilibrate when applied to a rigid body. 

Draw any triangle in each of these planes, then any force can be de 
composed into three components on the three sides of the triangle. Let 
x, y, z be the components of such a force in the first plane, and let x, y , z 
be the components of the corresponding force in the second plane ; we must 
then have equations of the form 

x = ax + by + cz, 
y = a x + b y + c z, 
z = a"x -f b"y + c"z, 

where a, b, c, &c., are constants. These equations do not contain any terms 
independent of the forces, because x, y , z must vanish when x, y, z vanish. 
They are linear in the components of the forces, because otherwise one force 
in one plane will not correspond uniquely with one force in the other. 

Let a? 1} y-i, z^ ac 2 , y 2 , z. 2 ; ... x n , y n , z n be the components of forces in the 
first plane. 

Let Xi, yi, Zi] #/, y 2 , z.\ ... x n , y n , z n be the components of the corre 
sponding forces in the second plane. Then we must have 

k = a-r k + by k + cz k , 
yk = a xk + b y k -f c z k , 
z k = a"x k +- b"y k + c"z k , 
where k has every value from 1 to n. If therefore we write 



and 



348] THE THEORY OF SCREW-CHAINS. 381 

with similar values for % % , 2z, 2/ then the above equations give 
2a? = a 2x + b % + c %z, 
% = a 2a? + 6 % + c/ ^z, 
2s = a"2a? + &"2y + c"2*. 

If the system of forces in the first plane equilibrate, the following con 
ditions must be satisfied : 

2tf = 0, % = 0, ^z = 0, 
and from the equations just written, these involve 

2a> =0, % =0, 2* =0, 
whence the corresponding system in the other plane must also equilibrate. 

To determine the correspondence it will be necessary to know only 
the three forces in the second plane which correspond to three given forces 
in the first plane. We shall then have the nine equations which will be 
sufficient to determine the nine quantities a, b, c, &c. 

It appears, from the form of the equations, that the ratio of the intensity 
of a force to the intensity of the corresponding force is independent of those 
intensities, i.e. it depends solely upon the situation of the lines in which the 
forces act. 

Take any four straight lines in one system, and let four forces, 
X 1} X 2 , X 3 , X^, on these four straight lines equilibrate. It is then well 
known that each of these forces must be proportional to certain functions 
of the positions of these straight lines. We express these functions by 
A ly A. 2 , A 3> A. The four corresponding forces will be X^, X 2 t X 3 , X 4 , 
and as they must equilibrate, they must also be in the ratio of certain 
functions AJ, AJ, A 3 , A of the positions. 

We thus have the equations 

Xi _ X 2 _X 3 _ -A-4 

A! A A 3 AI 

Xi _ X 2 _ X 3 _ X 

A 1 A 2 A 3 A 

We can select the ratio of X l to X^ arbitrarily : for example, let this ratio 
be /A; then 



whence the ratio of X 3 to X. 2 is known. Similarly the ratio of the other 
intensities X 3 : X 3 , and X 4 : X t is known. And generally the ratio of every 
pair of corresponding forces will be determined. 



382 THE THEORY OF SCREWS. [348, 

It thus appears that four straight lines in one system may be chosen 
arbitrarily to correspond respectively with four straight lines in the other 
system ; and that one force being chosen on one of these straight lines in 
one system, the corresponding force may be chosen arbitrarily on the corre 
sponding straight line in the other system. This having been done, the 
relation between the two systems is completely defined. 

From the case of parallel projections in two planes it is easy to pass to 
the case which will serve our present purpose. Instead of the straight lines 
in the two planes we shall take screws in two ^-systems. Instead of the 
forces on the lines we may take either twists or wrenches on the screws. 
More generally it will be better to use Plucker s word " Dyname," which we 
have previously had occasion to employ ( 260) in the sense either of a twist 
or a wrench, or even a twist velocity. We shall thus have a Dyname in one 
system corresponding to a Dyname in the other. 

Let us suppose that a Dyname on a screw of one n-system corresponds 
uniquely to a different Dyname on a screw of another ?i-system. The two 
n-systems may be coincident but we shall treat of the general case. 

In the first place it can be shown that if any number of Dynames in 
the first system neutralize, their corresponding Dynames in the second 
system must also neutralize. Take n screws of reference in one system, 
and also n screws of reference in the corresponding system. Let 6 be the 
Dyname in one system which corresponds to (f> in the other ; 6 can be 
completely resolved into component Dynames of intensities 1} ... 6 n on the 
n screws of reference in the first system and in like manner <f> can be resolved 
into n components of intensities fa, ... <f> n on the screws of reference in the 
second system (n = < 6). 

From the fact that the relation between and <f> is of the one-to-one 
type the several components <j> l} ...<f> n are derived from 1 ,...0 n by n equations 
which may be written 



fa t = ( ? *1) 0, + ( W 2) ,... S 4 (nn) 6 n , 

in which (11), (12), &c. must be independent of both and <, for otherwise 
the correspondence would not be unique. 

If there be a number of Dynames in the first system the sums of the 
intensities of their components on the n screws of reference may be expressed 
as S#!, ... ^0 n respectively. In like manner the sums of the intensities of 
the components of their correspondents on the screws of reference of the 
second system may be represented by "fa, ... 2</> n respectively. We therefore 



349] THE THEORY OF SCREW-CHAINS. 383 

obtain the following equations by simply adding the equations just written 
for each separate -screw 



20 H = (1) 20! + (w2) 20, . . . + (nn) 20 n . 

If the Dynamos in the first system neutralize then their components on 
the screws of reference must vanish or 



But it is obvious from the equations just written that in this case 



and therefore the corresponding Dynames will also neutralize. 

Given n pairs of corresponding Dynames in the two systems, we obtain 
?i 2 linear equations which will be adequate to determine uniquely the ?i 2 
constants of the type (11), (12), &c. It is thus manifest that n given pairs 
of Dynames suffice to determine the Dyname in either system, corresponding 
to a given Dyname in the other. It is of course assumed that in this case 
the intensities of the two corresponding Dynames in each of the ?i-pairs are 
given as well as the screws on which they lie. 

349. Properties of this correspondence. 

To illustrate the distinction between this Dyname correspondence and 
the screw correspondence previously discussed, let us take the case of two 
cylindroids. We have already seen that, given any three pairs of corre 
sponding screws, the correspondence is then completely defined ( 343). 
Any fourth screw on one of the cylindroids will have its correspondent on 
the other immediately pointed out by the equality of two anharmonic ratios. 
The case of the Dyname correspondence is, however, different inasmuch as 
we require more than two pairs of corresponding Dynames on the two 
cylindroids, in order to completely define the correspondence. For any 
third Dyname 6 on one of the cylindroids can be resolved into two Dynames 
6 l and 2 on the two screws containing the given Dynames. These com 
ponents will determine the components ^^^ on the corresponding cylindroid, 
which being compounded, will give <f> the Dyname corresponding to 0. 

It is remarkable that two pairs of Dynames should establish the corre 
spondence as completely as three pairs of screws. But it will be observed 
that to be given a pair of corresponding screws on the two cylindroids is 
in reality only to be given one datum. For one of the screws may be 
chosen arbitrarily; and as the other only requires one parameter to fix it 



384 THE THEORY OF SCREWS. [349, 

on the cylindroid to which it is confined its specification merely gives a 
single datum. To be given a pair of corresponding Dynamos is, however, 
to be given really two data one of these is for the screws themselves as 
before, while the other is derived from the ratio of the amplitudes. Thus 
while three pairs of corresponding screws amount to three data, two pairs 
of corresponding Bynames amount to no less than four data ; the additional 
datum in this case enabling us to indicate the intensity of each correspondent 
as well as the screw on which it is situated. 

It can further be shown in the most general case of the correspondence 
of the Bynames in two w-systems that the number of pairs of Bynames 
required to define the correspondence is one less than the number of pairs 
of screws which would be required to define merely a screw correspondence 
in the same two w-systems. In an n -system a screw has n l disposable 
co-ordinates. To define the correspondence we require n + 1 pairs of screws. 
Of course those on the first system may have been chosen arbitrarily, so 
that the number of data required for the correspondence is 



A Byname in an w-system has n arbitrary data, viz., n 1 for the screw, 
and one for the intensity : hence when we are given n pairs of corresponding 
Bynames we have altogether n 2 data. We thus see that the n pairs of 
corresponding Bynames really contribute one more datum to the problem 
than do the n + l pairs of corresponding screws. The additional datum is 
applied in allotting the appropriate intensity to the sought Byname. 

We can then use either the n pair of Byname correspondents or the 
(n + l) pairs of screw correspondents. In previous articles we have used the 
latter; we shall now use the former. 

350. Freedom of the fifth order. 

In the higher orders of freedom the screw correspondence does not indeed 
afford quite so simple a means of constructing the several pairs of corre 
sponding screws as we obtain by the Byname correspondence. In two 
five-systems the correspondence is complete when we are given five Bynames 
in one and the corresponding five Bynames in the other. To find the 
Byname X in the second system, corresponding to any given Byname A 
in the first system, we proceed as follows: Decompose A into Bynames 
on the five screws which contain the five given Bynames on the first system. 
This is always possible, and the solution is unique. These components will 
correspond to determinate Bynames on the five corresponding screws : these 
Bynames compounded together will give the required Byname X both in 
intensity and position. 

In the general case where a mass-chain possesses freedom of the fifth 



350] THE THEORY OF SCREW-CHAINS. 385 

order we may, by trial, determine five screw-chains about which the system 
can be twisted. Each set of five homologous screws will determine a 
five-system. In this method of proceeding we need not pay any attention 
to the intermediate screws : it will only be necessary to inscribe one Dyname 
(in this case a twist) in each of the homologous five-systems so that 
the group of six shall be homographic. The set of twists so found will form 
a displacement which the system must be capable of receiving. This is 
perhaps the simplest geometrical presentment of which the question 
admits. 

One more illustration may be given. Suppose we have a series of planes, 
and three arbitrary forces in each plane. We insert in one of the planes 
any arbitrary force, and its parallel projection can then be placed in all 
the other planes. Suppose a mechanical system, containing as many distinct 
elements as there are planes, be so circumstanced that each element 
is free to accept a rotation about each of the three lines of force in the 
plane, and that the amplitude of the rotation is proportional to the intensity 
of the force ; it must then follow that the system will be also free to accept 
rotations about any other chain formed by an arbitrary force in one plane and 
its parallel projections in the rest. 

We may, however, also examine the case of a mass-chain with freedom 
of the fifth order by the aid of the screw correspondence without intro 
duction of the Dyname. We find, as before, five independent screw-chains 
which will completely define all the other movements which the system 
can accept. To construct the subsequent screw-chains, which are quadruply 
infinite in variety, we begin by first finding any sixth screw-chain of the 
system by actual composition of any two or more twists about two or more 
of the five screw-chains. When a sixth chain has been ascertained the 
construction of the rest is greatly simplified. Each set of six homologous 
screws lie in a five-system. Place in each of these five-systems another 
screw which, with the remaining six, form a set which is homographic with 
the corresponding set in each of the other five-systems. These screws 
so determined then form another screw-chain about which the system must 
be free to twist. 

In the choice of the first screw with which to commence the formation 
of any further screw-chains of the five-system we have only a single condition 
to comply with : the screw chosen must belong to a given five-system. 
This implies that the chosen screw must be reciprocal merely to one given 
screw. On any arbitrary cylindroid a screw can be chosen which is reciprocal 
to this screw, and consequently on any cylindroid one screw can always be 
selected wherewith to commence a screw-chain about which a mass-chain 
with freedom of the fifth order must be free to twist. 

B. 25 



386 THE THEORY OF SCREWS. [351, 

351. Freedom of the sixth order. 

In freedom of the sixth order we select at random six displacements of 
which the mass-chain admits, and then construct the six corresponding screw- 
chains. The homologous screws in this case lie on six-systems, but a six- 
system means of course every conceivable screw. It is easily shown ( 248) 
that if to one screw in space corresponds another screw, and conversely, then 
the homography is completely established when we are given seven screws 
in one system, and the corresponding seven screws in the other. Any eighth 
screw in the one system will then have its correspondent in the other imme 
diately determined. 

It is of special importance in the present theory to dwell on the type of 
homography with which we are here concerned. If on the one hand it 
seems embarrassing, from the large number of screws concerned, on the 
other hand we are to recollect that the question is free from the complication 
of regarding the screws as residing on particular n-systems. Seven screws 
may be drawn anywhere, and of any pitch ; seven other screws may also be 
chosen anywhere, and of any pitch. If these two groups be made to corre 
spond in pairs, then any other screw being given, its corresponding screw will 
be completely determined. Nor is there in this correspondence any other 
condition, save the simple one, that to one screw of one system one screw 
of the other shall correspond linearly. 

Six screw-chains having been found, a seventh is to be constructed. 
This being done, the construction of as many screw-chains as may be desired 
is immediately feasible. From the homographic relations just referred to 
we have appropriate to each element of the system seven homologous screws, 
and also appropriate to each consecutive pair of elements we have the seven 
homologous intermediate screws. An eighth screw, appropriate to any 
element, may be drawn arbitrarily, and the corresponding screw being con 
structed on each of the other systems gives at once another screw-chain about 
which the system must be free to twist. 

When a mass-chain has freedom of the sixth order we see that any one 
element may be twisted about any arbitrary chosen screw, but that the 
screw about which every other element twists is then determined, and so 
are also the ratios of the amplitudes of the twists, by the aid of the inter 
mediate screws. 

352. Freedom of the seventh order. 

Passing from the case of six degrees of freedom to the case of seven 
degrees, we have a somewhat remarkable departure from the phenomena 
shown by the lower degrees of freedom. Give to the mass-chain any seven 
arbitrary displacements, and construct the seven screw-chains, a, /3, 7, B, e, 



352] THE THEORY OF SCREW-CHAINS. 387 

t) by twists about which those displacements could have been effected. 

In the construction of an eighth chain, 0, we may proceed as follows : 

Choose any arbitrary screw lm Decompose a twist on 1 into components 
on !, &, ry 1( $ lt GI , . This must be possible, because a twist about any 
screw can be decomposed into twists about six arbitrary screws, for we 
shall not discuss the special exception when the six screws belong to a system 
of lower order. 

The twists on a^ ... , &c., determine the twists on the screw-chains 
,... and therefore the twists on the screws a 2 , ... ,, which compound 
into a twist on <9 2 , similarly for 3 , &c.; consequently a screw-chain of which 
#! is the first screw, and which belongs to the system, has been constructed. 
This is, however, only one of a number of screw-chains belonging to the 
system which have 6 l for their first screw. The twist on l might have been 
decomposed on the six screws, &, 7l , S lt l , , 77,, and then the screws 2 , &c., 
might have been found as before. These will of course not be identical with 
the corresponding screws found previously. Or if we take the whole seven 
screws, o^, ... 77!, we can decompose a twist on 1 in an infinite number of 
ways on these seven screws. We may, in fact, choose the amplitude of the 
twist on any one of the screws of reference, oq , for example, arbitrarily, and 
then the amplitudes on all the rest will be determined. It thus appears 
that where l is given, the screw 2 is not determined in the case of freedom 
of the seventh order ; it is only indicated to be any screw whatever of a 
singly infinite number. The locus of 2 is therefore a ruled surface ; so will 
be the locus of 3 , &c. and we have, in the first place, to prove that all these 
ruled surfaces are cylindroids. 

Take three twists on 1} such that the arithmetic sum of their amplitudes 
is zero, and which consequently neutralize. Decompose the first of these 
into twists on a 1} &, 7l , S l} , ^, the second on a,, ft, 7l , 8 1} e ,, 17,, and the 
third on a 1} ft, %, 8 ly e 1} . It is still open to make another supposition 
about the twists on 6^ let us suppose that they are such as to make the two 
components on ^ vanish. It must then follow that the total twists on each 
of the remaining six screws, viz. 1} ft, 7l) 8^ e ly shall vanish, for their 
resultant cannot otherwise be zero. All the amplitudes of the twists about 
the screw-chains of reference must vanish, and so must also the amplitudes 
of the resultant twists when compounded. We should have three different 
screws for 2 corresponding to the three different twists on 0^ and as the 
twists on these screws must neutralize, the three screws must be co- 
cylindroidal. 

We can, therefore, in constructing a screw-chain of this system, not only 
choose #! arbitrarily, but we can then take for 6 2 any screw on a certain 
cylindroid : this being done, the rest of the screw-chain is fixed, including 
the intermediate screws. 

252 



388 THE THEORY OF SCREWS. [353, 

353. Freedom of the eighth and higher orders. 

If the freedom be of the eighth order, then it is easily shown that the 
first screw of any other chain may be taken arbitrarily, and that even the 
second screw may be chosen arbitrarily from a three-system. Passing on to 
the twelfth order of freedom, the two first screws of the chain, as well as the 
amplitudes of their twists, may be chosen quite arbitrarily, and the rest of 
the chain is fixed. In the thirteenth order of freedom we can take the 
two first twists arbitrarily, while the third may be chosen anywhere on a 
cylindroid. It will not now be difficult to trace the progress of the chain 
to that unrestrained freedom it will enjoy when the mass-chain has G^ 
degrees of freedom, when it is able to accept any displacement whatever. In 
the last stage, prior to that of absolute freedom, the system will have its 
position defined by 6/j, - 1 co-ordinates. A screw-chain can then be chosen 
which is perfectly arbitrary in every respect, save that one of its screws must 
be reciprocal to a given screw. 

354. Reciprocal Screw-Chains. 

We have hitherto been engaged with the discussion of the geometrical 
or kinematical relations of a mass-chain of p elements : we now proceed to 
the dynamical considerations which arise when the action of forces is 
considered. 

Each element of the mass-chain may be acted upon by one or more 
external forces, in addition to the internal forces which arise from the reaction 
of constraints. This group of forces must constitute a wrench appropriated 
to the particular element. For each element we thus have a certain wrench, 
and the entire action of the forces on the mass-chain is to be represented by 
a series of /j, wrenches. Recalling our definition of a screw-chain, it will be 
easy to assign a meaning to the expression, wrench on a screw-chain. By this 
we denote a series of wrenches on the screws of the chain, and the ratio of 
two consecutive intensities is given by the intermediate screw, as before. 
We thus have the general statement : 

The action of any system of forces on a mass-chain may be represented 
by a wrench on a screw-chain. 

Two or more wrenches on screw-chains will compound into one wrench 
on a screw-chain, and the laws of the composition are exactly the same as 
for the composition of twists, already discussed. 

Take, for example, any four wrenches on four screw-chains. Each set of 
four homologous screws will determine a four-system; the resulting wrench- 
chain will consist of a series of wrenches on these four-systems, each being 
the "parallel projection" of the other. 



354] THE THEORY OF SCREW-CHAINS. 389 

Let a and /3 be two screw-chains, each consisting of ^ screws, appro 
priated one by one to the p elements of the mass-chain. If the system 
receive a twist about the screw-chain /3, while a wrench acts on the screw- 
chain a, some work will usually be lost or gained ; if, however, no work be 
lost or gained, then the same will be true of a twist around a acting on 
a wrench on /3. In this case the screw-chains are said to be reciprocal. The 
relation may be expressed somewhat differently, as follows : 

If a mass-chain, only free to twist about the screw-chain a, be in equilibrium, 
notwithstanding the presence of a wrench on the screw-chain j3, then, conversely, 
a mass-chain only free to twist about the screw-chain ft will be in equilibrium, 
notwithstanding the presence of a wrench on the screw-chain a. 

This remarkable property of two screw-chains is very readily proved 
from the property of two reciprocal screws, of which property, indeed, it 
is only an extension. 

Let ! . . . a M be the screws of one screw-chain, and /3] . . . yS^ those of the 
other. Let a^, a. 2 , ... a M denote amplitudes of twists on a 1} 2 , &c., and let 
a i"> 82", &c., denote the intensities of wrenches on a 1} a 2 , &c. Then, from 
the nature of the screw-chain, we must have 

/ : / = < : a 2 "=a3 : a 8 ", &c., 
A : A" = & :&" = & :&", &c.; 

for as twists and wrenches are compounded by the same rules, the inter 
mediate screws of the chain require that the ratio of two consecutive 
amplitudes of the twists about the chain shall coincide with the ratio of the 
intensities of the two corresponding wrenches. Denoting the virtual 
coefficient of c^ and j3i by the symbol Br ai /3 lJ we have for the work done by 
a twist about a, against the screw-chain /3, 

&c., 



while for the work done by a twist about ft against the screw-chain a we 

have the expression 

2a 1 // A / OTaiPl + 2a 2 ^>a^ 2) &c. 

If the first of these expressions vanishes, then the second will vanish also. 

It will now be obvious that a great part of the Theory of Screws may be 
applied to the more general conceptions of screw-chains. The following 
theorem can be proved by the same argument used in the case when only a 
single pair of screws are involved. 

If a screw-chain 6 be reciprocal to two screw-chains a and /3, then 6 will 
be reciprocal to every screw-chain of the system obtained by compounding 
twists on a. and /3. 



390 THE THEORY OF SCREWS. [354, 

A screw-chain is defined by 6/4 1 data ( 353). It follows that a finite 
number of screw-chains can be determined, which shall be reciprocal to 
6/i 1 given screw-chains. It is, however, easy to prove that that number 
must be one. If two chains could be found to fulfil this condition, then 
every chain formed from the system by composition of two twists thereon 
would fulfil the same condition. Hence we have the important result 

One screw-chain can always be determined which is reciprocal to 6//- 1 
given screw-chains. 

This is of course only the generalization of the fundamental proposition 
with respect to a single rigid body, that one screw can always be found 
which is reciprocal to five given screws ( 25). 

355. Twists on 0/^ + 1 screw-chains. 

Given 6/^+1 screw-chains, it is always possible to determine the ampli 
tudes of certain twists about those chains, such that if those twists be 
applied in succession to a mass-chain of /u, elements, the mass-chain shall, 
after the last twist, have resumed the same position which it had before the 
first. To prove this it is first necessary to show that from the system formed 
by composition of twists about two screw-chains, one screw-chain can always 
be found which is reciprocal to any given screw-chain. This is indeed the 
generalization of the statement that one screw can always be found on a 
cylindroid which is reciprocal to a given screw. The proof of the more 
general theorem is equally easy. The number of screw-chains produced by 
composition of twists about the screw-chains a and /3 is singly infinite. 
There can, therefore, be a finite number of screw-chains of this system 
reciprocal to a given screw-chain 6. But that number must be one ; for if 
even two screw-chains of the system were reciprocal to 6, then every screw- 
chain of the system must also be reciprocal to 8. The solution of the original 
problem is then as follows : Let a. and /3 be two of the given 6/z, + 1 chains, 
and let 6 be the one screw-chain which is reciprocal to the remaining 
6/i 1 chains. Since the 6/* + 1 twists are to neutralize, the total quantity 
of work done against any wrench-chain must be zero. Take, then, any 
wrench-chain on 6. Since this is reciprocal to 6/4 1 of the screw-chains, the 
twists about these screw-chains can do no work against a twist on 6. It 
follows that the amplitudes of the twists about a and @ must be such that 
the total amount of work done must be zero. For this to be the case, the 
two twists on a and /3 must compound into one twist on the screw-chain v, 
which belongs to the system (a/3), and is also reciprocal to 0. This defines 
the ratio of the amplitudes of the two twists on a and /3. We may in fact 
draw any cylindroid containing three homologous screws of a, /3, and 7, then 
the ratio of the sines of the angles into which 7 divides the angle between 
a and @ is the ratio of the amplitudes of the twists on a. and /8. In a 



355] THE THEORY OF SCREW-CHAINS. 391 

similar manner the ratio of the amplitudes of any other pair of twists can 
be found, and thus the whole problem has been solved. 

We are now able to decompose any given twist or wrench on a screw - 
chain into 6//. components on any arbitrary 6//, chains. The amplitudes or 
the intensities of these G/A components may be termed the 6/i co-ordinates 
of the given twist or wrench. If the amplitude or the intensity be regarded 
as unity, then the 6/A quantities may be taken to represent the co-ordinates 
of the screw-chain. In this case only the ratios of the co-ordinates are of 
consequence. 

If the mass-chain have only n degrees of freedom where n is less than 6/1, 
then all the screw-chains about which the mass-chain can be twisted are so 
connected together, that if any n + 1 of these chains be taken arbitrarily, the 
system can receive twists about these n 4- 1 chains of such a kind, that after 
the last twist the system has resumed the same position which it had before 
the first. In this case n co-ordinates will be sufficient to express the twist 
or wrench which the system can receive, and n co-ordinates, whereof only the 
ratios are concerned, will be sufficient to define any screw-chain about which 
the system can be twisted. 

G/j, n screw-chains are taken, each of which is reciprocal to n screw- 
chains about which a mass-chain with freedom of the nth order can twist. 
The two groups of n screw-chains on the one hand, and 6/i - n on the other, 
may each be made the basis of a system of chains about which a mechanism 
could twist with freedom of the nth order, or of the (6/i - ?i)th order, re 
spectively. These two systems are so related that each screw-chain in the 
one system is reciprocal to all the screw-chains in the other. They may thus 
be called two reciprocal systems of screw-chains. 

Whatever be the constraints by which the freedom is hampered, the 
reaction of the constraints upon the elements must constitute a wrench on 
a screw-chain. It is a fundamental point of the present theory that this 
screw-chain belongs to the reciprocal system. For, as no work is done 
against the constraints by any displacement which is compatible with the 
freedom of the mass-chain, it must follow, from the definition, that the 
wrench-chain which represents the reactions must be reciprocal to all 
possible displacement chains, and must therefore belong to the reciprocal 
system. 

For a wrench-chain applied to the mass-chain to be in equilibrium it 
must, if not counteracted by some other external wrench-chain, be counter 
acted by the reaction of the constraints. Thus we learn that 



392 THE THEORY OF SCREWS. [355, 

Of two reciprocal screw-chain systems, each expresses the collection of 
ivrench-chains of which each one will equilibrate when applied to a mass-chain 
only free to twist about all the chains of the other system. 

This is, perhaps, one of the most comprehensive theorems on Equilibrium 
which could be enunciated. 

356. Impulsive screw-chains and instantaneous screw-chains. 

Up to the present we have been occupied with considerations involving 
kinematics and statics : we now show how the principles of kinetics can be 
illustrated by the theory sketched in this chapter. 

We shall suppose, as before, that the mechanical arrangement which we 
call the mass-chain consists of jj, elements, and that those elements are so 
connected together that the mass-chain has n degrees of freedom. We shall 
also suppose that the mass-chain is acted upon by a wrench about any screw- 
chain whatever. The first step to be taken is to show that the given 
wrench-chain may be replaced by another more conveniently circumstanced. 
Take any n chains of the given system, and 6/u, n chains of the reciprocal 
system, then the given wrench-chain can be generally decomposed into 
components on the n + (6/z - n) chains here specified. The latter, being all 
capable of neutralization by the reaction of the constraints, may be omitted, 
while the former n wrench-chains admit of being compounded into a single 
wrench-chain. We hence have the following important proposition : 

Whatever be the forces which act on a mass-chain, their effect is in general 
equivalent to that of a wrench on a screw-chain which belongs to the system of 
screw-chains expressing the freedom of the mass-chain. 

The application of this theorem is found in the fact that, while we still 
retain the most perfect generality, it is only necessary, either for twists or 
wrenches, to consider the system, defined by n chains, about which the mass- 
chain can be twisted. 

Let us consider the mass-chain at rest in a specified position, and suppose 
it receives the impulsive action of any set of forces, it is required to determine 
the instantaneous motion which the system will acquire. The first operation 
is to combine all the forces into a wrench-chain, and then to transform that 
wrench-chain, in the manner just explained, into an equivalent wrench- 
chain on one of the screws of the system. Let 6 be the screw-chain of the 
system so found. In consequence of this impulsive action the mass-chain, 
previously supposed to be at rest, will commence to move ; that motion can, 
however, be nothing else than an instantaneous twist velocity about a screw- 
chain a. We thus have an impulsive screw-chain 6 corresponding to an 
instantaneous screw-chain a. In the same way we shall have the impulsive 
screw-chains <, i/r, &c., correlated to the instantaneous chains, /3, y, &c. 



356] THE THEORY OF SCREW-CHAINS. 393 

The first point to be noticed is, that the correspondence is unique. To 
the instantaneous chain a one impulsive screw-chain 6 will correspond. There 
could not be two screw-chains 6 and 6 which correspond to the same instan 
taneous screw-chain a. For, suppose this were the case, then the twist 
velocity imparted by the impulsive wrench on 6 could be neutralized by the 
impulsive wrench on 6 . We thus have the mass-chain remaining at rest 
in spite of the impulsive wrenches on 6 and 6 . These two wrenches must 
therefore neutralize, and as, by hypothesis, they are on different screw- 
chains, this can only be accomplished by the aid of the reactions of the 
constraints. We therefore find that 6 and 6 must compound into a 
wrench-chain which is neutralized by the reactions of the constraints. 
This is, however, impossible, for 6 and 6 can only compound into a wrench 
on a screw-chain of the original system, while all the reactions of the 
constraints form wrenches on the chains of the wholly distinct reciprocal 
system. 

We therefore see that to each instantaneous screw-chain a only one 
impulsive screw-chain 6 will correspond. It is still easier to show that to 
each impulsive screw-chain 6 only one instantaneous screw-chain a will 
correspond. Suppose that there were two screw-chains, a and a. , either of 
which would correspond to an impulsive wrench on 6. We could then give 
the mass-chain, first, an impulsive wrench on d of intensity X, and make 
the mass-chain twist about a, and we could simultaneously give it an im 
pulsive wrench on the same screw-chain 6 of intensity X, and make the 
mass-chain twist about a . The two impulses would neutralize, so that as 
a matter of fact the mass-chain received no impulse whatever, but the 
two twist velocities could not destroy, as they are on different screw-chains. 
We would thus have a twist velocity produced without any expenditure of 
energy. 

We have thus shown that in the w-system of screw-chains expressing the 
freedom of the mass-chain, one screw-chain, regarded as an instantaneous 
screw-chain, will correspond to one screw-chain, regarded as an impulsive 
screw-chain, and conversely, and therefore linear relations between the 
co-ordinates are immediately suggested. That there are such relations can be 
easily proved directly from the laws of motion (see Appendix, note 7). We 
therefore have established a case of screw-chain homography between the 
two systems, so that if 1} ...0 n denote the co-ordinates of the impulsive 
screw-chain, and if a a , ... a n denote the co-ordinates of the corresponding 
instantaneous screw-chain, we must have n equations of the type 

0, = (11) ttl + (12) a 2 + (13) 3 . .. + (1 ) a B> 



(nn) B , 



394 THE THEORY OF SCREWS. [356, 

where (11), (12), &c., are n? coefficients depending on the distribution of the 
masses, and the other circumstances of the mass-chain and its constraints. 
The equations having this form, the necessary one-to-one correspondence is 
manifestly observed. 

357. The principal screw-chains of Inertia. 

We are now in a position to obtain a result of no little interest. Just as 
we have two double points in two homographic rows on a line, so we have n 
double chains in the two homographic chain systems. If we make, in the 
foregoing equations, 

0i = p*i 5 #2 = p 2 , &c., 

we obtain, by elimination of a 1} ... a n , an equation of the nth degree in p. 
The roots of this equation are n in number, and each root substituted in the 
equations will enable the co-ordinates of each of the n double screw-chains 
to be discovered. The mechanical property of these double chains is to be 
found in the following statement : 

If any mass-chain have n degrees of freedom, then in general n screw- 
chains can always be found (but not more than n), such that if the mass-chain 
receive an impulsive wrench from any one of these screw-chains, it will 
immediately commence to move by twisting about the same screw-chain. 

In the case where the mass-chain reduces to a single rigid body, free or 
constrained, the n screw-chains to which we have just been conducted reduce 
to what we have called the n principal screws of inertia. In the case, still 
more specialized, of a rigid body only free to rotate around a point, the 
theorem degenerates to the well-known property of the principal axes. We 
may thus regard the n principal chains now found as the generalization of 
the familiar property of the principal axes for any system anyhow con 
strained. 

Considerable simplification is introduced into the equations when, instead 
of choosing the chains of reference arbitrarily, we select the n principal 
screw-chains for this purpose ; we then have the very simple results, 

0, = (11) x ; 2 = (22) 2 ; ... 6 n = (nn) a n . 

This gives a method of finding the impulsive screw-chain corresponding to 
any instantaneous screw-chain. It is only necessary to multiply the co 
ordinates of the instantaneous screw-chain i, a 2 by the constant factors (11), 
(12), &c., in order to find the co-ordinates of the impulsive screw-chain. 

The general type of homography here indicated has to be somewhat 
specialized for the case of impulsive screw-chains and instantaneous screw- 
chains. The n double screw-chains are generally quite unconnected we 
might, indeed, have exhibited the relation between the two homographic 



357] THE THEORY OF SCREW-CHAINS. 395 

systems of screw-chains by choosing n screws quite arbitrarily as the double 
screws of the two systems, and then appropriating to them n factors (11), 
(22), (33), &c., also chosen arbitrarily. In the case of impulsive and instant 
aneous chains, the n double chains are connected together by the relation 
that each pair of them are reciprocal, so that the whole group of n chains 
form what may be called a set of co-reciprocals. 

To establish this we may employ some methods other than those 
previously used. Let us take a set of w-co-reciprocal chains, and let the 
co-ordinates of any other two chains, 6 and < of the same system, be 1> ... 6 n 
and fa, ...fa,,. Let 2p 1} *2p 2 , &c., 2p n be certain constant parameters 
appropriated to the screws of reference. 2pj is, for example, the work done 
by a twist of unit amplitude on the first screw-chain of reference against a 
wrench of unit intensity on the same chain. The work done by a twist O l 
against a wrench fa on this chain is 2p 1 1 fa. As the chains of reference are 
co-reciprocal, the twist on 6 l does no work against the wrenches fa 2 , < 3 , ... &c. ; 
hence the total work done by a twist on 6 against the wrench on </> is 



and hence if 6 and < be reciprocal, 



The quantities p 1 , . . . p n are linear magnitudes, and they bear to screw-chains 
the same relation which the pitches bear to screws. If we use the word 
pitch to signify half the work done by a unit twist on a screw-chain against 
the unit wrench on the screw-chain, then we have for the pitch p 6 of the 
chain 6 the expression 



The kinetic energy of the mass-chain, when animated by a twist velocity of 
given amount, depends on the instantaneous screw-chain about which the 
system is twisting. It is proportional to a certain quadratic function of the n 
co-ordinates of the instantaneous screw. By suitable choice of the screw- 
chains of reference it is possible, in an infinite number of ways, to exhibit 
this function as the sum of n squares. It follows from the theory of 
linear transformations that it is generally possible to make one selection of 
the screw-chains of reference which, besides giving the energy function the 
required form, will also exhibit p e as the sum of n squares. This latter 
condition means that the screw-chains of reference are co-reciprocal. It only 
remains to show that the n screw-chains of reference thus ascertained must 
be the n principal screw-chains to which we were previously conducted. 

We may show this most conveniently by the aid of Lagrange s equations 
of motion in generalized co-ordinates ( 86). 



396 THE THEORY OF SCREWS. [357- 

Let #j , . . . 6 n represent the co-ordinates of the impulsive screw-chain, and 
let !,... a n be the co-ordinates of the corresponding instantaneous screw- 
chain, reference being made to the screw-chains of reference just found. 

Lagrange s equations have the form 

A ( dT .\ -dT- p 
dt \ddj cfaj 

where T is the kinetic energy, and where PiSc^ denotes the work done 
against the forces by a twist of amplitude BO.I. 

If Q" denote the intensity of the impulsive wrench, then its component 
on the first screw of reference is f "0 1} and the work done is 2p 1 "0 l &a l , 
while, since the chains are co-reciprocal, the work done by Sttj against the 
components of &" on the other chains of reference is zero, we therefore have 

JV*%$r r fc 

We have also 

T=M(u 1 2 d 1 *+...+u n *d n 2 ), 

when u lt ... u n are certain constants. 

We have, therefore, from Lagrange s equation, 



whence, integrating during the small time t, during which the impulsive 
force acts, 



in which d is the actual twist velocity about the screw-chain, so that d l = dx l , 
each being merely the expression for the component of that twist velocity 
about the screw-chain. 

We hence obtain lt ... n , proportional respectively to 



Pi " Pn 



u 2 

If we make = (11), &c., we have the previous result, 
PI 



n = (nn)a n . 

358. Conjugate screw-chains of Inertia. 

From the results just obtained, which relate of course only to the chains of 
reference, we can deduce a very remarkable property connecting instantaneous 
chains, and impulsive chains in general. Let a. and ft be two instantaneous 
chains, and let and $ be the two corresponding impulsive chains, then when OL 



359] THE THEORY OF SCREW-CHAINS. 397 

is reciprocal to </>, ft will be reciprocal to 6. This, it will be observed, is a 
generalization of a property of which much use has been previousl)<jfcnade 
( 81). The proof is as follows. 

The co-ordinates of the instantaneous chains are 

&.-.. 

The co-ordinates of the corresponding impulsive chains are 



Pi Pn 

and 

V& Ujfin 

Pi Pn 

If the chain a be reciprocal to the impulsive chain which produces /3, then 
we have 

<!& + ... + u n 2 nl3 n = ; 

but this being symmetrical in a and /3 is precisely the same as the condition 
that the impulsive chain corresponding to a. shall be reciprocal to /9. Following 
the analogy of our previous language we may describe two screw-chains so 
related as conjugate screw-chains of Inertia. 

359. Harmonic screw-chains. 

We make one more application of the theory of screw-chains to the 
discussion of a kinetical problem. Let us suppose that we have any material 
system with n degrees of freedom in a position of stable equilibrium under 
the action of a conservative system of forces. If the system receive a small 
displacement, the forces will no longer equilibrate, but the system will be 
exposed to the action of a wrench on a screw-chain. We thus have two 
corresponding sets of screw-chains, one set being the chains about which the 
system is displaced, the other set for the wrenches which are evoked in 
consequence of the displacements. 

By similar reasoning to that which we have already used, it can be shown 
that these two corresponding chain systems are homographic. We can 
therefore find n screw-chains about which, if the system be displaced, a 
wrench will be evoked on the same screw-chain, and (the forces having a 
potential) it can be shown that this set of n screw-chains are co-reciprocal. 

If after displacement the system be released it will continue to make 
small oscillations. The nature of these oscillations can be completely 
exhibited by the screw-chains. To a chain a, regarded as an instantaneous 
screw-chain, will correspond the screw as an impulsive screw-chain. To the 
chain a, regarded as the seat of a displacing twist, will correspond a wrench 



398 THE THEORY OF SCREWS. [359 

(f) which is evoked by the action of the forces. It will, of course, generally 
happen that the chain 6 is different from the chain <. It can however be 
shown that 6 and </> are not in every case distinct. There are n different 
screw-chains, each of which regarded as a will have the two corresponding 
screws 6 and < identical. Nor is it difficult to see what the effect of such a 
displacement must be on the small oscillations which follow. A wrench is 
evoked by the displacement, and since 6 and < coincide, that wrench is 
undistinguishable from an infinitely small impulsive wrench which will 
make the system commence to twist about a. We are thus led to the 
result that 

There are n screw-chains such that if the system be displaced by a twist 
about one of these screw-chains, and then released, it will continue for ever to 
twist to and fro on the same screw-chain. 

Following the language previously used, we may speak of these as 
harmonic screw-chains, and it can be shown that whatever be the small 
displacement of the system, and whatever be the small initial velocities with 
which it is started, the small oscillations are merely compounded of twist 
vibrations about the n harmonic screw-chains. 



CHAPTER XXV. 

THE THEORY OF PERMANENT SCREWS*. 

360. Introduction. 

In commencing this chapter it will be convenient to recite a well-known 
dynamical proposition, and then to enlarge its enunciation by successive 
abandonment of restrictions. 

Suppose a rigid body free to rotate around a fixed point. There are, as 
is well known, three rectangular axes about any one of which the body 
when once set in rotation will continue to rotate uniformly so long as the 
application of force is withheld. These axes are known as permanent axes. 
The freedom of the body in this case is of a particular nature, included in 
the more general type known as Freedom of the Third Order. The Freedom 
of the Third Order is itself merely one subdivision of the class which, 
including the six orders of freedom, embraces every conceivable form of 
constraint that can be applied to a rigid body. We propose to investigate 
what may be called the theory of permanent screws for a body constrained 
in the most general manner. 

The movement of the body at each moment must be a twist velocity 
about some one screw belonging to the system of screws prescribed by 
the character of the constraints. In the absence of forces external to those 
arising from the reactions of the constraints, the movement will not, in 
general, persist as a twist about the same screw 6. The instantaneous screw 
will usually shift its position so as to occupy a series of consecutive positions 
in the system. It must, however, be always possible to compel the body to 
remain twisting about 6. For this purpose a wrench of suitable intensity 
on an appropriate screw 77 may have to be applied. Without sacrifice of 
generality we can in general arrange that 77 is one of the system of screws 

* Trans. Roy. Irish Acad., Vol. xxix. p. 613 (1890). 



400 THE THEORY OF SCREWS. [360- 

which expresses the freedom of the body ( 96). It may sometimes appear 
that the intensity of the necessary wrench on 77 vanishes. The body in 
such a case requires no coercion beyond that of the original constraints to 
preserve 9 as the screw about which it twists, and when this is the case we 
shall describe as a permanent screw. This use of the word permanent 
does not imply that the body could remain for ever twisting about this 
screw, for the movement of the body to an appreciable distance will in 
general entail some change in its relation to the constraints. The character 
istic of the permanent screw is the absence of any acceleration in the body 
twisting about it, using the word acceleration in its widest sense. 

In the earlier parts of the chapter we shall discard the restrictions 
involved in the assumption that the material arrangement is only a single 
rigid body. The doctrine of screw-chains (Chap, xxiv.) enables us to extend 
a considerable portion of the present theory to any mass-chain whatever. 
Any number of material parts connected in any manner must still conform 
to the general law, that the instantaneous movements can always be repre 
sented by a twist about a certain screw-chain. In general the mass-chain 
will have a tendency to wander from twisting about the original screw-chain. 
In such cases the position of the instantaneous screw-chain cannot be 
maintained without the imposition of further coercion than that which the 
constraints supply. This additional set of forces may be applied by a 
restraining wrench-chain, the relation of which to the instantaneous screw- 
chain we shall have to consider. Sometimes it may appear that no restraining 
wrench-chain is necessary beyond one of those provided by the reaction of 
the constraints. The instantaneous screw-chain is then to be described as 
permanent. 

361. Different properties of a Principal Axis. 

Another preliminary matter should be also noticed, because it exhibits 
the relation of the subject discussed in this chapter to some other parts of 
the Theory of Screws. In the ordinary theory of the rigid body there are, 
as is well known, two distinct properties of a principal axis which possess 
dynamical significance. We may think of a principal axis as the axis of a 
couple which, when applied impulsively to the body, will set it rotating 
about this axis. We may also think of the principal axis as a direction 
about which, if a body be once set in rotation, it will continue to rotate. 
The first of these properties by suitable generalization opens up the theory 
of principal screw-chains of inertia, which we have already explained in 
previous chapters. It is from the other property of the principal axis that 
the present investigation takes its rise. It is important to note that two 
quite different departments in the Theory of Screws happen to coalesce in 
the very special case of a rigid body rotating around a point. 



362] THE THEORY OF PERMANENT SCREWS. 401 

362. A Property of the Kinetic Energy of a System. 

It is obvious that the mere alteration of the azimuth about a fixed axis 
from which a rigid body is set into rotation will not affect its kinetic energy, 
provided the position of the axis and the angular velocity both remain un 
altered. 

A moment s reflection will show that this principle may be extended to 
any movement whatever of a rigid body. At each instant the body is 
twisting about some instantaneous screw a with a twist velocity a. Let 
the body be stopped in a position which we call A. Let it receive a dis 
placement by a twist of any amplitude about a and thus be brought to a 
position which we call B*. Finally, let the body be started from its new 
position B so as to twist again about a with the original twist velocity d, 
then it is plain that the kinetic energy of the body just before being stopped 
at the position A is the same as its kinetic energy just after it is started 
from the position B. 

Enunciated in a still more general form the same principle is as 
follows : 

Any mass-chain in movement is necessarily twisting about some screw- 
chain. If we arrest the movement, displace the mass-chain to an adjacent 
position on the same screw-chain, and then start the mass-chain to twist 
again on the same screw-chain, with its original twist velocity, the kinetic 
energy must remain the same as it was before the interruption. 

This principle requires that whatever be the symbols employed, the 
function T, which denotes the kinetic energy, must satisfy a certain identical 
equation. I propose to investigate this equation, and its character will 
perhaps be best understood by first discussing the question with co-ordinates 
of a perfectly general type. We shall suppose the mass-chain has n degrees 
of freedom. 

Let the co-ordinates x lt ...x n represent the position of the mass-chain, 
and let its instantaneous motion be indicated by x l ,...x n . Let be the 
initial position of the mass-chain, then in the time Bt it has reached the 
position , whereof the co-ordinates are 

x l + x^t, ... x n + x n $t. 

The movement from to must, like every possible movement of a 
system, consist of a twist about a screw-chain. This is a kinematical fact, 
wholly apart from whatever particular system of co-ordinates may have 

* We have supposed that the pitch of this displacement is the same as the pitch of a. This 
restriction is only introduced here because the constraints will generally forbid the body to make 
any other twist about the axis of a. If the body were quite free we might discard the restriction 
altogether as is in fact done later on ( 376). 

B. 26 



402 THE THEORY OF SCREWS. [362, 

been adopted. We call this screw-chain 0, and 6 denotes the twist velocity 
with which the system moves round 0. 

Choose any n independent screw-chains, about each one of which the 
mass-chain is capable of twisting. Then 6 can be decomposed into n com 
ponents which have twist velocities 1} ... 6 n about the several screw-chains 
of reference ( 355). 

Since everything pertaining to the position or the movement of the 
mass-chain must necessarily admit of being expressed in terms of the co 
ordinates of the mass-chain, and since the quantities 1} ... O n are definitely 
determined by the position and movements of the mass-chain, it follows 
that these quantities must be given by a group of formulae which may be 
written 



"n Jn\%i y H>n j #1 > Xnj- 

Let the mass-chain be stopped in the position A. Let it then be dis 
placed to an adjacent position B defined by the following variations of the 
co-ordinates 



where 8e is a small quantity. From this new position B let the mass-chain 
be started into motion so that it shall have the same twist velocity as it 
had just before being stopped in A and about the same screw- chain. This 
condition requires that each of the quantities lt ... Q n shall resume its original 
value unaltered by the stoppage and subsequent restarting of the mass- 
chain from a new position. There must accordingly be an adjustment of 
8x l} ... 8x n to satisfy the equations 



dx n 



Under these circumstances T has obviously not altered, so that we 
have also 

dT dT . dT .. dT 



Let us assume, for brevity, the symbol A, such that 

* _ . d d 

dx l dx n 



363] 



THE THEORY OF PERMANENT SCREWS. 



403 



Then we obtain, by elimination of Bx l} ... 8x n , and 8e, 



AT, 



AA, 



dT dT 



df n dfn 



= 0. 



dx l " dx 1t 

Such is the general condition which must be satisfied by the kinetic energy 
of any material arrangement whatever. But the equation is so complicated 
when expressed in ordinary rectangular co-ordinates that there is but little 
inducement to discuss it. 

363. The Identical Equation in Screw-chain Co-ordinates. 

The Theory of Screw-chains exhibits this equation in a form of special 
simplicity. For, suppose that 



then the equation of the last article reduces to 



We thus have the following theorem : 

If the co-ordinates, #/ , ... O n , of a mass-chain be n twists about n screiv- 
chains, belonging to the system of screw-chains which express the freedom of 
the mass-chain, and if Oi, ... O n be the twist velocities of the mass-chain about 
these same screw-chains, then the kinetic energy T satisfies the equation 



A dT 



v i ~jm~> f Un ja~ f ~ 
a PJ au n 

I have thought it instructive to exhibit the origin of this equation as 
a special deduction from the case of co-ordinates of the general type. For 
a brief demonstration the following simple argument suffices : 

If the mass-chain be displaced through S0/, ... S0 n while the velocities 
are unaltered, the change of kinetic energy is 



If the change of the position be due to a small twist Se around the screw- 
chain with co-ordinates l} ... 6 n , then 



262 



404 THE THEORY OF SCREWS. [363- 

but from the physical property of the kinetic energy already cited, it appears 
that this kind of displacement cannot change the kinetic energy, whence 

A dT , dT _ 

" n ~ 



364. The Converse Theorem. 

Let us take the general case where the co-ordinates are oc l ,...x n and 
x l , ... x n . Suppose that cb 2 , ... x n are all zero, then x^ is the velocity of the 
mass-chain. We shall also take x z , ... x n to be zero, so that we only consider 
the position of the mass-chain defined by x l . Think now of the two positions 
for which ^ = and x 1 = x 1 respectively. Whatever be the character of the 
constraints it must be possible for the mass-chain to pass from the position 
X-L = to the position x l #/ by a twist about a screw-chain. The magnitude 
Xi is thus correlated to the position of the mass-chain on a screw-chain about 
which it twists. 

If the co-ordinates are of such a kind that the identical equation which T 
must necessarily satisfy has the form 

. dT . dT _ 

Xl dxJ~ Xn dx n ~ 

then for the particular displacement corresponding to the first co-ordinate, 
# 2 , ... x n are all zero, and 



and as T must involve x^ in the second degree, we have 

T = Hx* 

where H is independent of #/ 

Let #j be the twist velocity about the screw-chain corresponding to the 
first co-ordinate, then, of course, A being a constant, 

T = Ae*, 
whence A 6f = Hxf, 

VZtfj = A/774, 
and by integration and adjustment of units and origins 

& . 

We thus see that while the displacement corresponding to the first co-ordinate 
must always be a twist about a screw-chain, whatever be the actual nature of 
the metric element chosen for the co-ordinate, yet that when the identical 
equation assumes the form 



366] THE THEORY OF PERMANENT SCREWS. 405 

the metric element must be essentially the amplitude of the twist about the 
screw-chain. We have thus proved the following theorem : 

The co-ordinates must be twists about n screw-chains of reference whenever 
the identical equation, satisfied by T, assumes the form 

6 dT . a dT_ 
9l " ffn ~ 



365. Transformation of the Vanishing Emanant. 

Suppose that the position and movement of a mass-chain were represented 
by the co-ordinates #/, #</, n ; #i> @2, - &n when referred to one set of 
n screws of reference, and by </>/, (f> 2 } ... <,/; </>!, < 2 , </> when referred to 
another set of screws of reference. Then of course these sets of co-ordinates 
must be linearly connected. 

We may write 

4 - (11) ft .-.. + (!)*, , 

= (nl) </... + (>,)<,;. 

Then, by differentiation 

4 -(ii) ^... 



Thus the two sets of variables are co-gredients, and by the theory of 
linear transformations we must have 

dT , dT dT : dT 



The expression on either side of the equation is of course known in algebra 
as an emanant ( 261). 

We could have foreseen this result from the fact that whatever set of 
n independent screw-chains belonging to the system was chosen, the identical 
equation must in each case assume the standard form. 

366. The General Equations of Motion with Screw- chain 
Co-ordinates. 

The screw-chain co-ordinates of a mass-chain with n degrees of freedom 
are #/, ... d n ; the co-ordinates of the velocities are 1} ... 0*. Let 77 be the 
wrench-chain which acts on the system. Let the components of the wrench- 
chain, when resolved on the screw-chains of reference, have for intensities 
Vi > W- Let p l , ... p n be the pitches of the chains of reference, by which 
is meant that 2p t is the work done on that screw-chain by a twist of unit 
amplitude against a wrench of unit intensity on the same screw-chain. Then 
the screw-chains of reference being supposed to be co-reciprocal, we have, 



406 THE THEORY OF SCREWS. [366- 

irom Lagrange s equations, 

d (dT\ dT 

~ = ^W.T?] . 

dt \d0J dd, w - 



dt\de 

These equations admit of a transformation by the aid of the identity 

e^- f dT -o 
*ap" n de n - 

Differentiating this equation by & lt we find 



but 



dT * _ , 

** 



= Q _ _ , 

dt \dej ~ J dd? 2 de.de, n de, de n * de^ei + n de~de~f 

whence, by substitution 

AfdT.} = 0fT d * T dT 

dt\dej ~ l de^" n 



Hence when screw-chain co-ordinates are employed Lagrange s equations 
may be written in the form 



ifT d T d T 

* 



367. Generalization of the Eulerian Equations. 

The equations just written can be further simplified by appropriate 
choice of the screw-chains of reference. We have already assumed the 
screw-chains of reference to be co-reciprocal. If, however, we select that 
particular group which forms the principal screw-chains of inertia ( 357), 
then every pair are conjugate screw-chains of inertia besides being reciprocal. 
In this case T takes the form 



368] THE THEORY OF PERMANENT SCREWS. 407 

Neglecting the small quantities 6-^ ... &c. we have 



-r = 0, &C. 



d6.de, 
Introducing these values we obtain 



dT 



These may be regarded as the generalization for any material arrangement 
whatever of the well-known Eulerian equations for the rotation of a rigid 
body around a fixed point. If there are no external forces then i\" , . . . rj n " 
are all zero, and the equations of movement assume the simple form 

AT 






368. The Restraining Wrench-chain. 

If a mass-chain be twisting about an instantaneous screw-chain 0, the 
mass-chain will, in general, presently forsake 6 and gradually adopt one 
instantaneous screw-chain after another. It is however possible, by the 
application of a suitable wrench-chain, to compel the mass-chain to continue 
twisting about the same screw 6 with unchanged twist velocity. We now 
proceed to the discovery of this restraining wrench-chain when no other 
external forces act on the mass- chain. 

As all the accelerations of 6 must vanish, the co-ordinates of the wrench- 
chain required are obtained by imposing the conditions 

1 = 0; a = 0,...0 n =0. 

We therefore infer from the general equations of 366, that if IJ L ", . . . r) n " 
are the co-ordinates of the restraining wrench-chain we must have 

1 dT 



1 dT 



408 THE THEORY OF SCREWS. [368- 

whcnce we deduce the following theorem : 

If the position of a mass-chain be referred to co-reciprocal screw-chains 
of reference, then 

1 dT 1 dT 

Pi dffi p n d6n 

are the co-ordinates of the restraining wrench- chain which would coerce the 
mass-chain into continuing to twist about the same screw-chain 6*. 

369. Physical meaning of the Vanishing Emanant. 

We may verify this theorem by the following method of viewing the 
subject. It must be possible to coerce the system to twist about 6 by the 
imposition of special constraints. The reactions of these constraints will 
constitute, in fact, the restraining wrench-chain. It is, however, a character 
istic feature that, as the system is, ex hypothesi, still at liberty to twist 
about 6, the reaction of any constraints which are consistent with this 
freedom must lie on a screw-chain reciprocal to 0. 

The condition that two screw-chains, 6 and rj, shall be reciprocal ( 354) 



s 



but this is clearly satisfied if for i) 1} ... we substitute 

l^dT^ 1 dT 

piW p n d0 n ] 

for the equation then becomes 

dT i. fl dT n 

i ddr" n de~r 

which, when multiplied by 0, reduces to the known identity 

e dT ^ dT o 

*ap"- + *n?--?.- 

We thus obtain a physical meaning of this equation. It is no more than an 
expression of the fact that the restraining wrench-chain must be reciprocal 
to the instantaneous screw-chain. 

370. A displacement without change of Energy. 

It should also be noticed that provided the twist velocities remain un 
altered the kinetic energy will be unchanged by any small displacement 
of the mass-chain arising from a twist on any screw-chain reciprocal to the 
restraining screw-chain. 

* A particular case of this, or what is equivalent thereto, is given in Williamson and Tarleton s 
Dynamics, 2nd ed., p. 432. 



372] THE THEORY OF PERMANENT SCREWS. 409 

For, if 6-^, ... d,i be the co-ordinates of the displacement, the change in 
Tis 

^.+ +0 ~ 

which may be written 

O dT- i0 L<W-. 

but this will be zero if, and only if, the screw-chain #/, ... n be reciprocal 
to the screw-chain 

^ dT ^dT^ 

p l ddj " p n dQn 

371. The Accelerating Screw-chain. 

When the mass-chain has forsaken the instantaneous screw-chain 6, and 
is twisting about another instantaneous screw-chain <, there must be a 
twist velocity about some screw-chain p, which, when compounded with the 
twist velocity about 6, gives the twist velocity about </>. When < and are 
indefinitely close, then p is the accelerating screw-chain. 

Taking the n principal screw-chains of inertia as the screws of reference 
and assuming that external forces are absent, we have 

,i dT 

11 = dl 7 



,if 20 - 
a u n u n 



It is plain that the co-ordinates of the accelerating screw-chain are 1 , ... n , 
whence we have the following theorem: 

If a mass-chain be twisting around a screw-chain 6, and if external forces 
are absent, the co-ordinates of the corresponding accelerating screw-chain are 

proportional to 

I dT I dT 



372. Another Proof. 

It is known from the theory of screw-chains (357) that if a quiescent 
mass-chain receive an impulsive wrench-chain with co-ordinates 

11 2 11 2 

til U-n 

TT^ 1 " pn> 

P\ Pn 

the mass-chain commences to twist about the screw-chain, of which the 
co-ordinates are 

pi, Pn- 



410 THE THEORY OF SCREWS. [372- 

Jf, by imposition of a restraining wrench-chain, the mass-chain continues 
to twist about the same screw-chain 6, the restraining wrench-chain has 
neutralized the acceleration. It follows that the restraining wrench-chain, 
regarded as impulsive, must have generated an instantaneous twist velocity 
on the accelerating screw-chain, equal and opposite to the acceleration that 
would otherwise have taken place. The co-ordinates of this impulsive 
wrench-chain are proportional to 

dT 1 dT 

Pl d0 l " " Pnd0 n " 

The corresponding instantaneous screw-chain is obtained by multiplying 
these expressions severally by 



and thus we find, as before, for the co-ordinates of the accelerating screw- 
chain 



373. Accelerating Screw-chain and instantaneous Screw-chain. 

We have, from the expressions already given, 

. dT rlT 

M(utf& + ...+ utfjj = e, JL + . . . e n j* , . 

But the right-hand side is the emanant which we know to be zero, whence 



This shows that 1} ... 6 n , and lf ... d n are on conjugate screw-chains of 
inertia, and hence we deduce the following theorem : 

Whenever a mass-chain is moving without the action of external forces, 
other than from constraints restricting the freedom, the instantaneous screw- 
chain and the accelerating screw-chain are conjugate screw-chains of inertia. 

374. Permanent Screw-chains. 

Reverting to the general system of equations ( 366) we shall now in 
vestigate the condition that 6 may be a permanent screw-chain. It is obvious 
that if 0J, ... n are all zero, then 

dT dT_ 
d0 i " " d0 n 

must each be zero. If, conversely, the differential coefficients just written 
are all zero, then the quantities j , ... n must each also vanish. 



375] THE THEORY OF PERMANENT SCREWS. 411 

This is obviously true unless it were possible for the determinant 



dd? 



d*T 



to become zero. Remembering that T is a homogeneous function of the 
quantities l) ... n in the second degree, the evanescence of the determinant 
just written would indicate that T admitted of expression by means of n 1 
square terms, such as 



This vanishes if 



= 0, &c.; 



each of these is a linear equation in lt ... n , and consequently a real system 
of values for 6^ ... n must satisfy these equations, and render T zero. It 
would thus appear that a real motion of the mass-chain would have to be 
compatible with a state of zero kinetic energy. This is, of course, im 
possible ; it therefore follows that the determinant must not vanish, and 
consequently we have the following theorem : 

If the screw-chains of reference be co-reciprocal, then the necessary and 
the sufficient conditions for 6 to be a permanent screw are that its co-ordinates 
l , 2 , ... n shall satisfy the equations 

dT =o- ^=o 

dOl d0 n 

There are n of these equations, but they are not independent. The cmanant 
identity shows that if n 1 of them be satisfied, the co-ordinates so found 
must, in general, satisfy the last equation also. 

375. Conditions of a permanent Screw-chain. 

As the quantities #/, . . . tl are small, we may generally expand T in 
powers, as follows : 

T=T + 1 T 1 +...0 n T n 



The equation 
therefore becomes 



dT 



412 THE THEORY OF SCREWS. [375, 

and as #/, a , &c., are indefinitely small, this reduces to 

2*1 = 0, 
where 2\ is a homogeneous function of lt 2 , ... B n in the second degree. 

For the study of the permanent screws we have, therefore, n equations 
of the second degree in the co-ordinates of the instantaneous screw-chain, 
and any screw-chain will be permanent if its co-ordinates render the several 
differential coefficients zero. We may write the necessary conditions that 
have to be fulfilled, as follows : 

Let us denote the several differential coefficients of T with respect to the 
variables by I, II, III, &c. Then the emanant identity is 

1 i + 2 n + 3 ni + ...=o, 

and we may develop any single expression, such as III, in the following 
form : 

III = III! A 2 + III 22 2 2 + III^s 2 + 2111,-M + . . . 2III M 4 . 
As the emanant is to vanish identically, we must have the coefficients of 
the several terms, such as Of, Q?Q. ly A0Ai & c -> all zero, the result being 
three types of equation 

In = 0, I-B+ II 12 = 0, I 23 +II 1 3 + IIIi 2 = > 

1122 = 0, II U + I 12 =0, &C., 

11133 = 0, II B + III B = > &C. f 

IV^ = 0, &c., &c. 

&c. 
Of the first of these classes of equations, I n = 0, there are n, of the second 

A e^ +u- A n(n-l)(n-2) . w(n+l)(n+2) 

there are n (n - 1), and of the third, ^^ , in all, - i ^ 5 . 

1 . L . o 1 . Z 

376. Another identical equation. 

Let T be the kinetic energy of a perfectly free rigid body twisting for 
the moment around a screw 6. It is obvious that T will be a function of 
the six co-ordinates, #/, . . . # 6 , which express the position of the body, and 
also of 0j, ... 6> the co-ordinates of the twist velocity, 

T=f(0 l ,...0. t ft, ...A). 

We may now make a further application of the principle employed in 
302. The kinetic energy will be unaltered if the motion of the body be 
arrested, and if, after having received a displacement by a twist of amplitude 
e about a screw of any pitch on the same axis as the instantaneous screw, 
the body be again set in motion about the original screw with the original 
twist velocity. This obvious property is now to be stated analytically. 



376] THE THEORY OF PERMANENT SCREWS. 413 

It has been shown in 265 that if lt 2 , ... 6 are the co-ordinates of 
a screw of zero pitch, then the co-ordinates of a screw of pitch p x on the 
same axis are respectively 

* p x dR A p x dR * p x dR 

"l~l ---- - ~ , t/2 "1 -- -- T~ , . . . t/u T -- ~ - . 

^ d0 1 4p 2 d0 2 4>p 6 d0 G 

In these expressions p x denotes an arbitrary pitch, while R is the 
function 

es + fa ... + 6 2 + 2 (12) eA + 2 (23) e,e 3 , &c., 

where (12) is the cosine of the angle between the first and second screws of 
reference, and similarly for (23), &c. 

The principle just stated asserts that T must remain unchanged if we 
substitute for #/, #2 , & c - the expressions 



We thus obtain the formula 

p^dR^dT , A p,dRy^ = Q 

^dejde, 4>p 6 deJ d0 6 

As this must be true for every value of p x , we must have, besides the 
vanishing ernanant, the condition 

^dR dT^ 1^?^ =0 

p l dO, dffi p 6 d0 6 d0 6 ~ 

It is plain that this is equivalent to the statement that the screw whose 
co-ordinates are 

l^dR T_dR ^dR 
~Pi d0\ P* d0 2 p* d6 6 

must be reciprocal to the screw defined by the co-ordinates 

I dT_ l^dT^ \dT 

Pl d0 l " p,d0:r " p 6 d0 6 " 

The former denotes a screw of infinite pitch parallel to and hence it 
follows that the restraining screw must be perpendicular to 6. Remember 
ing also that the restraining screw is reciprocal to 0, it follows that the 
restraining screw must intersect 0. We thus obtain the following result : 

If be the screw about which a free rigid body is twisting, then to check 
the tendency of the body to depart from twisting about a restraining wrench 
on a screw which intersect* at right angles must in general be applied. 



414 THE THEORY OF SCREWS. [377, 

377. Different Screws on the same axis. 

Let the body be displaced from a standard position to another position 
denned by the co-ordinates #/, 2 , . . . 6 , and let it then be set in rotation 
about a screw of zero pitch with a twist velocity whose co-ordinates are 
lt 2 , ... 6 . Let the kinetic energy of the body in this condition be T. 

Suppose that in addition to the rotation about 6 the body of mass M 
also received a velocity v of translation parallel to 0. Then the kinetic 
energy of the body would be T , where 



It is obvious that the position of the body, i.e. the co-ordinates $/, 0.f, ... e , 
can have no concern in ^Mv 2 , whence 



dT dT 
, a , = -J0-, , and similar equations. 

at/i CtC/! 

But a body rotating about with an angular velocity and translated 
parallel to with the velocity v is really rotating about a screw on the 
same axis as and with a pitch v -f- p. As v may have any value we obtain 
the following theorem : 

All instantaneous screws lying on the same axis have the same restraining 
screw. 

378. Co-ordinates of the Restraining Wrench for a free rigid 
body. 

Suppose the body to have a standard position from which we displace it 
by small twists #/, . . . 6 around the six principal screws of inertia. While 
the body is in its new position it receives a twist velocity of which the 
co-ordinates relatively to the six principal screws of inertia are lt ... 6 . 

To compute the kinetic energy we proceed as follows : Let a point lie 
initially at oc, y, z, then, by the placing of the body at the starting position 
the point is moved to X, Y, Z, where 

X = a (0, - 2 ) + y (0 5 f + 6 ) - z (0 3 f + 0/) + x, 
Y = b (0 3 f - 0/) + z (0, + 2 ) - x (0, + U ) + y, 
Z = c (0 r ; - 6 ) + x (0, + 0/) - y (0/ + a ) + z, 

in which a, b, c are the radii of gyration on the principal axes. The six 
principal screws of inertia lie, of course, two by two on each of the three 
principal axes, with pitches + a, a on the first, + b, b on the second, 
and + c, c on the third. 

In consequence of the twist velocity with the components #1,... 6 , each 
point X, Y, Z receives a velocity of which the components are 



378] THE THEORY OF PERMANENT SCREWS. 415 

a (6, - 0.) + Y(0 r , + e G )-Z (0 3 + 4 ), 

6 (B, -0J + Z (0! + 2 ) - X (e r> + 6 ), 

c (A - 0) + X (A + 4 ) - I 7 (9, + 4). 

Before substitution for X, Y, Z it will be convenient to use certain abbre 
viations, 

0i 2 = 1 ; &i 2 = pi ; 0i + 0. 2 = \i , BI + 2 = &)] , 

/]//}/ /} A Q i Q \ Q \ Q 

"3 #4 = e 2 , 3 (7 4 = p 2 , "3 + #4 = A-2- U 3 + (7 4 = <W 2 , 

$> # = e 3 ; 5 $ 6 = p 3 ; #5 + #/ = X 3 , 5 + fi = <w s - 

With these substitutions in v 2 the square of the velocity of the element we 
readily obtain after integration and a few reductions and taking the total 
mass as unity, 



2 c 2 ) 

. 2 (i)i bc^p^cos X 2 <w 3 &)j (c 2 a 2 ) 
> 3 &)i XsO^a^ (a 2 6 2 ), 
whence we easily find 

7/TT 



If T;/ , ... rj s " be the co-ordinates of the restraining wrench, then, as shown 
in 368, 

"= 1^ T 

P*d0r 

whence we deduce the following fundamental expressions for the co 
ordinates of the restraining wrench : 

pMi" = acp 3 w 2 + abp 2 w s + (6 2 c 2 ) &) 2 &) 3 , 
PM-i = + acp 3 (o. 2 abp 2 co 3 + (b" c 2 ) &> 2 &> 3 , 
PMs" = abpico 3 + cbpsWi + (c 2 a 2 ) w^w^ 
PPl" = + abpitos cbpsW! + (c 2 a 2 ) o) s o) l , 
PS^S" = bcpzW! + acpiO) 2 + (a 2 6 2 ) w^w*, 
Pets" = + fop^ acpi(o 2 + (a 2 6 2 ) 

As usual, we here write for symmetry 
p l= = + a; p., = -a; p 3 = + b; 

We verify at once that 

Pw"0 1 + ... 

but this is of course known otherwise to be true, because the restraining 
screw must be reciprocal to the instantaneous screw. 



416 THE THEORY OF SCREWS. [378- 

These equations enable us to study the correspondence between each 
instantaneous screw 6 and the corresponding restraining screw 77. It is to 
be noted that this correspondence is not of the homographic, or one-to-one 
type, such as we meet with in the study of the Principal Screws of Inertia, 
and in other parts of the Theory of Screws. The correspondence now to 
be considered has a different character. 

379. Limitation to the position of the Restraining Screw. 

If a particular screw 6 be given, then no doubt, a corresponding screw r) 
is given definitely, but the converse is not true. If 77 be selected arbitrarily 
there will not in general be any possible 6. If, however, there be any one 0, 
then every screw on the same axis as 6 will also correspond to the same 77. 

From the equations in the last article we can eliminate the six quantities, 
6 l , . . . (i ; we can also write ??/ = i)"^, . . . rj n " = tj"r) n where 77" is the intensity 
of the restraining wrench and ^,,..1)^ the co-ordinates of the screw on 
which it acts. 

We have a (%" + V ) = 2abp 2 a) 3 2ac/> 3 <w 2 , 



2 c 2 r) 1 + 7; 2 , p. 2 p 3 
whence = 6 c , 

a 7?! 7/2 C0 2 W 3 

and from the two similar equations we obtain, by addition, 
b- j^ 77! + 7/ 2 c 2 a 2 773 + 774 a 2 6 2 % + 77,; _ 

a % ^2 b *73 - *?4 c 775-776 

It might at first have been supposed that any screw might be the possible 
residence of a restraining wrench, provided the corresponding instantaneous 
screw were fitly chosen. It should however be remembered that to each 
restraining screw corresponds a singly infinite number of possible instan 
taneous screws. As the choice of an instantaneous screw has five degrees 
of infinity, it was to be presumed that the restraining screws could only 
have four degrees of infinity, i.e. that the co-ordinates of a restraining screw 
must satisfy some equation, or, in other words, that they must belong to a 
screw system of the fifth order, as we have now shown them to do. 

380. A verification. 

We confirm the expression for the co-ordinates of 77 in the following manner. 
It has been shown ( 376) that so long as retains the same direction and 
situation, its pitch is immaterial so far as 77 is concerned. This might have 
been inferred from the consideration that a rigid body twisting about a 
screw has no tendency to depart from the screw in so far as its velocity of 
translation is concerned. It is the rotation which necessitates the restrain 
ing wrench if the motion is to be preserved about the same instantaneous 



381] THE THEORY OF PERMANENT SCREWS. 417 

screw. We ought, therefore, to find that the expressions for the co-ordinates 
of ?) remained unaltered if we substituted for 1} ... 6 , the co-ordinates of 
any other screw on the same straight line as 6. These are ( 47) 



04), 04 - y (0 3 



where H is arbitrary. 

Introducing these into the values for rj^, it becomes 

- aca) 2 (ft + -- a) 3 } + abw. A U% + -?- j + (& 2 - c 2 ) &> 2 &> 3 , 
y C .. / \ / 

from which .H" disappears, and the required result is proved. 

The restraining screw is always reciprocal to the instantaneous screw, 
and, consequently, if e be the angle between the two screws, and d their 
distance apart, 

(Pi + Pe) cos e - d sin e = 0. 

We have seen that this must be true for every value of p g , whence 

cos e = ; d = ; 

i.e. the two screws must intersect at right angles as we have otherwise shown 
in 376. 

This also appears from the formulae 

Vi + W = 26p 2 &> 3 2c/9 3 &> 2 , 
1)3" + -n" = 2c/o 8 a>i - 2a/> 1 o> 3 , 



multiplying respectively by &>,, &> 2 , eo s , and adding, we get 

til + 17s) (01 + #2) + (% + ^4> (03 + 4 ) + (^75 + 17) (05 + 6 ) = 0, 



which proves that rj and ^ are rectangular ; but we already know that they 
are reciprocal, and therefore they intersect at right angles. 

381. A Particular Case. 

The expressions for the restraining wrenches can be illustrated by taking 
as a particular case an instantaneous screw which passes through the centre 
of gravity. 

B. 27 



418 THE THEORY OF SCREWS. [381- 

The equations to the axis of the screw are 

ap l + 3/0)3 zo) 2 _ bp 2 + za>i #a) 3 _ c/3 3 + xa) 2 yw^ 

0>1 &>2 W 3 

If x, y, z are all simultaneously zero, then 

ap 1= bp 1 = cp3 

ft)! ft> 2 0> 3 

and these are, accordingly, the conditions that the instantaneous screw passes 
through the centre of gravity. 

With these substitutions the co-ordinates become 

p l7)l " = (6 2 - c 2 ) ft> 2 ft> 3 ; Pals" = (c 2 - a 2 ) 30i ; PM*" = (a 2 - & 2 ) iw 2 , 
PM 9 " = (& - c 2 ) &> 2 a>3 ; PM" = (c 2 - a 2 ) u,^ ; jp^" = (a 2 - 6 2 ) &) 1 a) 2 ; 
remembering that ^ = + a ; p 2 = - a, &c., we have 

% " + V =0; 7 7 / + 7 74 // = 0; V+W-0; 

but these are the conditions that the pitch of 77 shall be infinite ; in other 
words the restraining wrench is a couple, as should obviously be the case. 

From the equations already given, we can find the co-ordinates of the 
instantaneous screw in terms of those of the restraining screw. 

We have 

__ 

~ 



2 (6 2 - c 2 ) (c 2 - a 2 ) (a 2 - 6 2 ) 



,-j-f O TT . JJ 

and O>i = tl - 77 -fj: ] f>2 = = ti T~, T, 77V j ^3 " ~~7 77 /7\ 

a (r} 1 7/2 ) (773 ^4 j C ^7/5 f] 6 ) 

If we make 
then we have 



3 
"* 



- . _ 

G " 



2pT "2o6c 

In these expressions, /> fl is the pitch of 6, and is, of course, an indeterminate 
quantity. 

382. Remark on the General Case. 

If the freedom of a body be restricted, then any screw will be permanent, 
provided its restraining screw belong to the reciprocal system. For the body 



383] THE THEORY OF PERMANENT SCREWS. 419 

will not depart from the original instantaneous screw except by an accelera 
tion. This must be on a screw which stands to the restraining screw in the 
relation of instantaneous to impulsive, but in the case supposed these two 
screws are reciprocal, therefore they cannot be so related, and therefore there 
is no acceleration. 

There is little to be said as to the restraining wrench when the freedom 
is of the first order. Of course, in this case, as every movement of the body 
can only be a twist about the screw which prescribes its freedom, the 
restraining wrench is provided by the reactions of the constraints. It is 
only where the body has liberty to abandon its original instantaneous screw 
that the theory of the restraining wrench becomes significant. 

383. Two Degrees of Freedom. 

If a rigid body has two degrees of freedom, then it is free to twist about 
every screw on a certain cylindroid. If the body be set initially in motion 
by a twist velocity about some one screw on the surface, then, in general, 
it will not remain twisting about this screw. A movement will take place 
by which the instantaneous axis gradually comes into coincidence with 
other screws on the cylindroid. If we impose a suitable restraining wrench 
i), then of course can be maintained as the instantaneous screw; 77 is 
reciprocal to 6. It may be compounded with any reactions of the constraints 
of the system. Thus, given 6, there is an entire screw system of the fifth 
order, consisting of all possible screws reciprocal to 0, any one screw of which 
may be taken as the restrainer. Of this system there is one, but only one, 
which lies on the cylindroid itself. There are many advantages in taking it 
as the restraining wrench, and it entails no sacrifice of generality ; we there 
fore have the following statement : To each screw on the cylindroid, regarded 
as an instantaneous screw, will correspond one screw, also on the cylindroid, 
as a restraining screw. 

The position of this restraining screw is at 
once indicated by the property that it must be 
reciprocal to the instantaneous screw. If we 
employ the circular representation for the 
screws on the cylindroid (fig. 42), and if be 
the pole of the axis of pitch, then it is known 
that the extremities of any chord, such as IR 
drawn through 0, will correspond to two re 
ciprocal screws ( 58). If therefore / be the 
instantaneous screw, R must be the restraining 

screw. If a body free to twist about all the screws on the cylindroid be set 
in motion by a twist velocity about /. it will be possible, by a suitable wrench 

27-2 




4 20 THE THEORY OF SCREWS. [383- 

applied on the screw corresponding to R, to prohibit the body from changing 
its instantaneous screw. 

Let be the pole of the axis of inertia, then, if I A be a chord drawn 
through , the points / and A correspond to a pair of conjugate screws of 
inertia ( 135). It further appears that A is the instantaneous screw corre 
sponding to an impulsive wrench on R ( 140). Therefore the effect of the 
wrench on R when applied to control the body twisting about 7 is to com 
pound its movement with a nascent twist velocity about A. Therefore A 
must be the accelerating screw corresponding to /. We thus see that 

Of two conjugate screws of inertia, for a rigid body with two degrees of 
freedom, either is the accelerator for a body animated by a twist velocity about 
the other. 

384. Calculation of T. 

In the case of freedom of the second order we are enabled to obtain the 
form of T, from the fact that the emanant vanishes, that is, 



If we assume that T is a homogeneous function of the second degree in 
0i and 2 , the solution of this equation must be 

T = M 2 + zseA + Me, 2 + H (6>;0 2 - o&y + (0/0 2 - #/ e,) (A0, + M), 

in which L, S, M, H, A are constants. If we further suppose that 0, and #./ 
are so small that their squares may be neglected, then the term multiplied 
by H may be discarded, and we have 

T = M 2 
whence 



Thus, for the co-ordinates of the restraining screw, supposing the screws of 
reference to be reciprocal, we have 



from which it is evident that 



which is, of course, merely expressing the fact that 77 and 9 are reciprocal. 

385. Another method. 

It may be useful to show how the form of T, just obtained, can be derived 
from direct calculation. I merely set down here the steps of the work and 
the final result. 



THE THEORY OF PERMANENT SCREWS. 421 

Let us take any two screws on the cylindroid a and ft, and let their co 
ordinates, when referred to the absolute screws of inertia, be 

ls ... 6 , and &, ... /8. 

Then any other screw on the cylindroid, about which the body has been 
displaced by a twist, by components #/ on a. and 2 on ft will have, for co 
ordinates, 

^ *MV...%tf*M . 

and the screw about which the body is twisting, with a twi^t velocity 0, will 
have, for co-ordinates, 



It readily appears that, so far as the terms involving #/and #./are concerned, 
the kinetic energy is the expression 



where 

A = + be (! + 2 ) [( 3 - 6 ) (& - &) - (3 - 4> (A - &)] 

+ (6 2 - c 2 ) (a, + 04) (a, + a 6 ) (A + & 

+ ca (a g + 4 ) [(a, - a.,) (& - /3 6 ) - ( 5 - 6 ) (A - A)] 

+ (c 2 - a 2 ) ( B + 6 ) (! + 2 ) (/3 S 

+ a6 (a s + 6 ) [(a, - 4 ) (A - &) - ( ai - a 2 ) (/3 3 - /3 4 )] 

+ (a 2 - 6 2 ) (! + a a ) ( 3 + 4 ) (A 



= + be (& + /8 8 ) [( 5 - 6 ) 09, - A) - (a, - 4 ) (A - 

- (Z> 2 - c 2 ) (a : + a, 



ca 3 + ! - 2 / B - / 6 - 5 - 

- (c 2 - a 2 ) ( 3 + 4 ) (/S B 



- a,) (A - /3 2 ) - (i - 

- (a 2 - 6 2 ) (a, + .) (A + A) (A + A). 



386. The Permanent Screw. 

We now write the equations of motion for a body which has two degrees 
of freedom, and is unacted upon by any force, the screws of reference being 
the two principal screws of inertia. 

We have, from the general equations (367) 

M" 2/3 ^^ 

= 



422 THE THEORY OF SCREWS. 

Introducing the value just obtained for T, 



[386- 



There must be one screw on the cylindroid, for which 



This screw will have the accelerations ^ and 2 , both zero, and thus we have 
the following theorem : 

If a rigid body has two degrees of freedom, then, among the screws about 
which it is at liberty to twist, there is one, and in general only one, which has 
the property of a permanent screw. 

The existence of a single permanent screw in the case of freedom of the 
second order seems a noteworthy point. The analogy here ceases between 
the permanent screws and the principal screws of inertia. Of the latter 
there are two on the cylindroid ( 84). 

387. Geometrical Investigation. 

Let N (fig. 43) be the critical point on the circle which corresponds to 
the permanent screw ( 50). Let P be a screw 0, the twist velocity about 




Oi 



Fig. 43. 

which is 0. Let u e be a linear parameter appropriate to the screw 6, such 
that Mu g 2 6 2 is the kinetic energy. 

Let O x and 2 be the two screws of reference on the cylindroid and for 
convenience let the chord 00 Z be unity. Let the point Q correspond to 
another screw <f>, then from 57 



Ptolemy s theorem gives 

PQ6(f> = 2 <l $102- 

Now let < be the adjacent screw about which the body is twisting in a time 
8t after it was twisting about 6. 



388] THE THEORY OF PERMANENT SCREWS. 423 

Then 

01 = @i + OiOt, 
<j>2 ~ $2 + d z ot, 

whence 



which is, accordingly, the rate at which P will change its position. If we 
substitute for 0j and 6 2 their values already found in the last article, we 
obtain for the velocity of P the expression 



N being the position of the permanent screw, let p be the length of the 
chord P^, then the expression just written assumes the form 



where k is a constant. 

This expression illustrates the character of the screw corresponding to N. 
If p be zero, then the expression for this velocity vanishes. This means that 
P has no tendency to abandon N ; in other words, that the screw correspond 
ing to N is permanent. 

388. Another method. 

It is worth while to investigate the question from another point 
of view. 

Let us think of any cylindroid 8 placed quite arbitrarily with respect to 
the position of the rigid body. A certain restraining screw i) will corre 
spond to each screw 6 on S. As 6 moves over the cylindroid, so must the 
corresponding screw 17 describe some other ruled surface S . The two 
surfaces, S and 8 , will thus have two corresponding systems of screws, 
whereof every two correspondents are reciprocal. One screw can be dis 
covered on S , which is reciprocal, not alone to its corresponding 6, but to 
all the screws on the cylindroid. A wrench on this ij can be provided by the 
reactions of the constraints, and, consequently, the constraints will, in this 
case, arrest the tendency of the body to depart from 6 as the instantaneous 
screw. It follows that this particular 6 is the permanent screw. 

The actual calculation of the relations between 77 and the cylindroid is as 
follows : 

A set of forces applied to a rigid system has components X, T, Z at 
a point, and three corresponding moments F, G, H in the rectangular planes 
of reference. 



424 THE THEORY OF SCREWS. [388 

Let p be the pitch of the screw on which the wrench thus represented 
lies, and let a, y, z be the co-ordinates of any point on this screw. Then, 
in the plane of Z the moments of the forces are xY yX, and if to this be 
added pX, the whole must equal H. 

Thus we have the three equations, so well known in statics, 

F=pX+yZ -zY, 
G=pY+zX-xZ, 
H=pZ + xY -yX. 

The centrifugal acceleration on a point P is, of course, co 2 PH, where o> is 
the angular velocity, and PH the perpendicular let fall on the axis. The 
three components of this force are X y Y , Z , where 

X = ft> 2 sin 6 (x sin y cos 0}, 
Y = ft) 2 cos 6 (y cos 6 x sin 0), 
Z = ft) 2 0-rasin 2(9), 

and the three moments are F , G , H , where 

F = to 2 sin 9 (yz sin + xz cos 6 Zmy cos 6), 
G = ft) 2 cos 6 ( yz sin 6 xz cos 6 + Zmx sin 0), 
H = to 2 1(2/ 2 - 2 ) sin cos 6 + xy cos 20}. 

We are now to integrate these expressions over the entire mass, and we 
employ the following abbreviations ( 324) : 

jxdm = Mx jydm = My ; $zdm = Mz ; 



jxydm = M1 3 2 ; fxzdm = Ml.? ; jyzdm = Ml? ; 
X = fX dm , Y=fY dm; Z = jZ dm; 
F = fF dm; G=fG dm; H=JH dm; 
then, omitting the factor Ma?, we have 

X= + (a? sin 6 y cos 6) sin 0, 
Y= (x sin 6 7/ cos 6) cos 0, 
Z = Z Q m sin 20 ; 

F = + sin (I, 2 sin + I* cos 0) - 2my sin cos 0, 
G = - cos (^ sin (9 + , 2 cos 0) + 2mx sin cos 0, 
#= (/>i 2 - p*} sin cos + r- cos 20. 
We can easily verify that 



388] THE THEORY OF PERMANENT SCREWS. 425 

We now examine the points on the cylindroid intersected by the axis of 
the screw 

F=pX+yZ -zY, 

G=pY+zX-xZ, 
H=pZ + xY-yX. 

We write the equations of the cylindroid in the form 

x R cos <f> ; y = R sin < ; z = m sin2</>; 
then, eliminating p and R, and making 



V = FX+GY+HZ, 
we find, after a few reductions, 

tan 3 ^(YV-GU) + tan 2 <f> (X V - FU + 2mX U) 



This cubic corresponds, of course, to the three generators of the cylindroid 
which the ray intersects. 

If we put 



then the cubic becomes, by eliminating m, 



The factor Ftan^ + Z simply means that the restraining screw cuts the 
instantaneous screw at right angles. 

The two other screws in which 77 intersects the cylindroid are given by 
the equation 

(XYV-XGU) tan 2 + (XYV- FUY) = 0. 

These two screws are of equal pitch, and the value of the pitch is 

Pl (XYV-XGU)+p 2 (FUY-XYV} 
U(FY-GX) 

where p^ and p. 2 are the pitches of the two principal screws on the cylindroid. 
After a few reductions the expression becomes 

V (I* - x opl ) sin + (I* + y opa ) cos 6 
U x Q sin y cos 6 

This is the pitch of the two equal pitch screws on the cylindroid which rj 
intersects. If 77 is to be reciprocal to the cylindroid, then, of course, the 
pitch of 77 itself should be equal and opposite in value to this expression. 
Hence the permanent screw on the cylindroid is given by 
(li* - XopJ sin 6 + (7 2 2 + y p 2 ) cos 6 = 0. 



426 THE THEORY OF SCREWS. [388- 

We notice here the somewhat remarkable circumstance, that if 

/! 2 - ajojpi = 0> and I? + y*P* = 0, 
then all the screws on the cylindroid are permanent screws. 

It hence appears that if two screws on a cylindroid are permanent, then 
every screw on the cylindroid is permanent. 

389. Three Degrees of Freedom. 

Let us now specially consider the case of a rigid body which has freedom 
of the third order. On account of the evanescence of the emanant we have 

*dT * dT , dT _ 

*W 4 * W 4 *W? 

It is well known that if U, V, W be three conies whose equations submit to 
the condition 



those conies must have three common intersections. 
It therefore follows that the three equations 



must have three common screws. These are, of course, the permanent 
screws, and, accordingly, we have the theorem : 

A rigid system which has freedom of the third order has, in general, three 
permanent screws. 

There will be a special convenience in taking these three screws as the 
screws of reference. We shall use the plane representation of the three- 
system, and the equations of the conies will be 

AJA + BAOi + CAO^O, or 7=0, 

AAo s + BAOi + cAo, = o, F=O, 
AAh + BA^ + cAe^o, w=o-, 
but, as e l u + e 2 v+0 3 w = o, 

identically, we must have 

^ = 0; A, = Q; A 3 = 0; 
(7 1= 0; C 2 =0; B 3 =0; 

and also A : + B 2 + C 3 = 0. 

For symmetry we may write 



390] THE THEORY OF PERMANENT SCREWS. 427 

We thus find that when T is referred to the three permanent screws of the 
system, its expression must be 

T= aB* + 60 2 2 + c0 3 2 + ZfdA + %ff$A + %h0A 

+ (ji-v) 010 A + (v-\) 0M + (X - A*) S A- 

Let 77" be the intensity of any wrench acting on a screw 77 belonging to 
the system, and let 2-or lr , represent the virtual coefficient between 77 and the 
first of the three screws of reference. 

Then, substituting for T in Lagrange s equations, we have 
+ ah + h 2 + g 3 -(p-v) eA = -n-^7?", 



If 77 be the restraining screw, then an appropriate wrench ?/ should be 
capable of annihilating the acceleration, i. e. of rendering 

0j = ; 2 = ; S = ; 
whence the position of 77, and the intensity 77" are indicated by the equations 

(V - 



We can now exhibit the nature of the correspondence between 77 and 0, for 



If we make H=$i$A+n"> and omit the dots over lt &c., we have 



+ w 32 77 3 ) - H (a - 7), 
03 Oia*?! + ^ 32 i?2 +p 3 r ns) =H(@- a). 
We may reduce them to two homogeneous forms, viz. 



where L = - M=; N = \ 

d^ * ^772 ^773 

390. Geometrical Construction for the Permanent Screws. 

We see that 77 must lie on the polar of the point lt 2 , 3 with respect 
to the pitch conic ( 201) or the locus of all the screws for which 



428 THE THEORY OF SCREWS. [390, 

We also see that 77 must lie on the polar of the point a#i, /3# 2 > 7$s with 
regard to the same conic. 

We thus obtain a geometrical construction by which we discover the 
restraining screw when the instantaneous screw is given. 

Two homographic systems are first to be conceived. A point of the first 
system, of which the co-ordinates are 8 1} # 2 > &s, has as its correspondent a 
point in the second system, with co-ordinates a0 1} /30 2 , j0 3 . The three 
double points of the homography correspond, of course, to the permanent 
screws. 

To find the restraining screw ?? corresponding to a given instantaneous 
screw 0, we join to its homographic correspondent, and the pole of this 
ray, with respect to the pitch conic, is the position of 17. 

The pole of the same ray, with regard to the conic of inertia ( 211), is 
the accelerator. It seems hardly possible to have a more complete geo 
metrical picture of the relation between 77 and 9 than that which these 
theorems afford. 

391. Calculation of Permanent Screws in a Three-system. 

When a three-system is given which expresses the freedom of a body 
we have seen how in the plane representation the knowledge of a conic (the 
conic of inertia) will give the instantaneous screw corresponding to any 
given impulsive screw. A conic is however specified completely by five 
data. The rigid body has nine co-ordinates. It therefore follows that there 
is a quadruply infinite system of rigid bodies which with respect to a given 
three-system will have the same conic of inertia. If in that three-system a 
be the instantaneous screw corresponding to 77 as the impulsive screw for 
any one body of the quadruply infinite system, then will 77 and a stand in 
the same relation to each other for every body of the system. 

The point in question may be illustrated by taking the case of a four- 
system. The screws of such a system are represented by the points in space, 
and the equation obtained by equating the kinetic energy to zero indicates 
a quadric. For the specification of the quadric nine data are necessary. 
This is just the number of co-ordinates required for the specification of 
a rigid body. If therefore the inertia quadric in the space representation 
be assumed arbitrarily, then every instantaneous screw corresponding to a 
given impulsive screw will be determined; in this case there is only a finite 
number of rigid bodies and not an infinite system for which the correspond 
ence subsists. 

We thus note that there is a special character about the freedom of the 
fourth order which we may state more generally as follows. To establish 
a chiastic homography ( 292) in an n-system requires (n l)(n + 2)/2 data. 



391] THE THEORY OF PERMANENT SCREWS. 429 

If the restraints are such that the number of degrees of freedom is less 
than four, then an infinite number of rigid bodies can be designed, such 
that the impulsive screws and their corresponding instantaneous screws 
shall be represented by a given chiastic homography. If n exceed four then 
it will not in general be possible to design a rigid body such that its corre 
sponding impulsive screws and instantaneous screws shall agree with a given 
chiastic homography. If, however, n 4 then it is always possible to design 
one but only one rigid body so that its pairs of corresponding impulsive 
screws and instantaneous screws shall be represented by a given chiastic 
homography. 

Returning to the three-system we may remark that, having settled the 
inertia conic in the plane representation we are not at liberty to choose 
three arbitrary points as representing the three permanent screws. For 
if these three points were to be chosen quite arbitrarily, then six relations 
among the co-ordinates of the rigid body would be given, and the conic of 
inertia would require five more conditions. Hence the co-ordinates of the 
rigid body would have in general to satisfy eleven conditions which, of 
course, is not generally possible, as there are only nine such co-ordinates. 
It is therefore plain that when the conic of inertia has been chosen at least 
two other conditions must necessarily be fulfilled by the three points which 
are to represent the permanent screw. This fact is not brought out by 
the method of 389 in which, having chosen the three permanent screws 
arbitrarily, we have then written down the general equation of a conic 
as the inertia conic. This conic should certainly fulfil at least two con 
ditions which the equations as there given do not indicate. 

We therefore calculate directly the expression for the kinetic energy of 
a body in the position #/, # 2 , 3 twisting about a screw with twist velocities 
#1, 6 2 , 6 3 when the screws of reference are the three principal screws of the 
three-system with pitches a, b, c, and when a? , y , z , If, 1 2 2 , 1 3 2 , pf, p 2 * } p/ are 
the nine co-ordinates ( 324) of the rigid body relative to these axes. 

It is easily shown that we have for the kinetic energy the mass M of the 
body multiplied into the following expression where squares and higher 
powers of #/, # 2 > QS are omitted : 

i (a 2 ^ 2 + W + c 2 3 2 + p*0* + p*ej + p 3 *0 3 *) 

+ 0A (c -b)x + 0A (a -c)y + 0A (b - a) z. 



+ 0A(p* 2 - p>? + uc - ab) 
+ 0A (V - cz ) - 6 A (4 2 



430 



THE THEORY OF SCREWS. 

+ 0A (Pi 2 - p 3 2 + ba - be) 



+ 0A (/> 2 2 - pi +cb- GO) 
+ 0A (I? - fyo) - 3 0i (li 2 



[391- 



+ 0* 



+ 01 



The coefficients of #/, # 2 , 0s respectively each equated to zero will give 
three conies 7=0, F= 0, Tf = 0. These conies have three common points 
which are of course the three permanent screws. 

If we introduce a new quantity O we can write the three equations 



(I/ - 



+ (l*-ax*)0 2 + (ab - p-> + H) 3 = 0. 

The elimination of ft between each pair of these equations will produce 
the three equations U = Q, V0, W = 0. If therefore we eliminate 1} 2 , 3 
from the three equations just written the resulting determinant gives a 
cubic for H. The solution of this cubic will give three values for fl which 
substituted in the three equations will enable the corresponding values of 
0i > 0zy 0s to be found. We thus express the co-ordinates of the three per 
manent screws in terms of the nine co-ordinates of the rigid body and their 
determination is complete. 

It may be noted that the same permanent screws will be found for any 
one of the systems of rigid bodies whose co-ordinates are 



whatever h may be. 

392. Case of Two Degrees of Freedom. 

We have already shown that there is a single permanent screw in every 
case where the rigid body has two degrees of freedom. We can demonstrate 
this in a different manner as a deduction from the case of the three- 
system. 

Consider a cylindroid in a three-system, that is of course a straight 
line in the plane representation ( 200). Let this line be AB (tig. 44). If 
the movements of the body be limited to twists about the screws on the 
cylindroid, there may be reactions about the screw which corresponds to the 
pole P of this ray with respect to the pitch conic, in addition to the reactions 
of the three-system. 



393] THE THEORY OF PERMANENT SCREWS. 431 

The permanent screw on this cylindroid will be one whereof the restrain 
ing screw coincides with P. In gene.ral, the points corresponding homo- 




Fig. 44. 

graphically to the points on the ray AB will form a ray CD. The inter 
section 0, regarded as on CD, will be the correspondent of some point X on 
AB. The restraining screw corresponding to X will therefore lie at P, and 
will be provided by the constraints. Accordingly, X is a permanent screw 
on the cylindroid, and it is obvious from the construction that there can be 
no other screw of the same character. 

We can also deduce the expression for T in the two-system from the 
expression of the more general type in the three-system ; for we have 

T= T + o - v) o; e A +(v-\) o;d A + (*-/*) # 3 0A. 

Consider any screw on the cylindroid defined by 



substituting, we obtain 

(0, 8 - 2 3 ) [(X - ft) P0 2 + (\-v) Q9 3 ], 

which we already know to be the form of the function in the case of the two- 
system ( 384). 

393. Freedom of the Fourth Order. 

The permanent screws in the case of a rigid body which has freedom of 
the fourth order may be investigated in the following manner: If a screw 
be permanent, the corresponding restraining screw 77 must be provided by the 
reactions of the constraints. All the reactions in a case of freedom of the 
fourth order lie on the screws of a cylindroid. On a given cylindroid three 
possible 77 screws can be found. For, if we substitute a l + \^ 1 , 2 + X/3 2 , &c., 
for T/!, 7/2, &c., in the equation 



{ c-a [ 

b 7? 3 - 7/4 



432 THE THEORY OF SCREWS. [393-397 

we obtain a cubic for X. The three roots of this cubic correspond to three 77 
screws. Take the 6 corresponding to one of the 77 screws, then, of course, 
6 will not, in general, belong to the four-system. We can, however, assign 
to Q any pitch we like, and as it intersects 77 at right angles, it must cut two 
other screws of equal pitch on the cylindroid ( 22). Give to 6 a pitch equal 
and opposite to that of the two latter screws, then 6 is reciprocal to the 
cylindroid, and therefore it belongs to the four-system. We thus have a 
permanent screw of the system, and accordingly we obtain the following 
result : 

In the case of a rigid body with freedom of the fourth order there are, in 
general, three, and only three, permanent screws. 

394. Freedom of the Fifth and Sixth Orders. 

When a rigid body has freedom of the fifth order, the screws about which 
the body can be twisted are all reciprocal to a single screw p. In general, p 
does not lie on the system prescribed by the equation which the co-ordinates 
of all possible 77 screws have to satisfy. It is therefore, in general, not 
possible that the reaction of the constraints can provide an 77. There are, 
however, three screws in any five-system which possess the property of 
permanent screws without however making any demand on the reaction of 
the constraints. The existence of these screws is thus demonstrated : 
Through the centre of inertia of the body draw the three principal axes, 
then, on each of these axes one screw can always be found which is reciprocal 
to p. Each of these will belong to the five-system, and it is obvious from 
the property of the principal axes, that if the body be set twisting about one 
of these screws it will have no tendency to depart therefrom. 

A body which has freedom of the sixth order is perfectly free. Any screw 
on one of the principal axes through the centre of inertia is a permanent 
screw, and, consequently, there is in this case a triply infinite number of 
permanent screws. 

395. Summary. 

The results obtained show that for a rigid body with the several degrees 
of freedom the permanent screws are as follows : 

No. of Permanent Screws 



Freedom I 


1 


II. 


1 


Ill . 


3 


IV. . 


3 


V. 


3 



VI Triply infinite 



CHAPTER XXVI. 

AN INTRODUCTION TO THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 

396. Introduction. 

The Theory of Screws in non-Euclidian space is a natural growth from 
some remarkable researches of Clifford* in further development of the 
Theory of Riemann, Cayley, Klein and Lindemann. I here give the in 
vestigation sufficiently far to demonstrate two fundamental principles 
( 42 7> 434) which conduct the theory to a definite stage at which it 
seems convenient to bring this volume to a conclusion. 

I have thought it better to develop from the beginning the non- Euclidian 
geometry so far as we shall at present require it. It is thus hoped to make 
it intelligible to readers who have had no previous acquaintance with this 
subjectf. I give it as I have worked it out for my own instruction*. It is 
indeed characteristic of this fascinating theory that it may be surveyed from 
many different points of view. 

397. Preliminary notions. 

Let #!, # 2 , x 3 , x t be four numerical magnitudes of any description. We 
may regard these as the co-ordinates of an object. Let y lt y 2 , y 3t y 4 be the 
co-ordinates of another object, then we premise that the two objects will be 
identical if, and only if 

1 ^2 3 4 

?/l ~ #2 ~ 2/3 ~ 2/4 

All possible objects may be regarded as constituting a content. 

"Preliminary Sketch of Biquaternions," Proceedings of the London Mathematical Society, 
Vol. iv. 381395 (1873). See also "On the Theory of Screws in a Space of Constant Positive 
Curvature," Mathematical Papers, p. 402 (1876). Clifford s Theory was much extended by 
the labours of Buchheim and others ; see the Bibliographical notes. 

t We are fortunately now able to refer English readers to a Treatise in which the Theory of 
non-Euclidian space and allied subjects is presented in a comprehensive manner. Whitehead, 
Universal Algebra, Cambridge, 1898. 

Trans. Roy. Irish Acad., Vol. xxvm. p. 159 (1881), and Vol. xxix. p. 123 (1887). 
B - 28 



434 THE THEORY OF SCREWS. [397- 

All objects whose co-ordinates satisfy one linear homogeneous equation 
we shall speak of as an extent. 

All objects whose co-ordinates satisfy two linear homogeneous equations 
we shall speak of as a range. 

It must be noticed that the content, with its objects, ranges, and extents, 
have no necessary connexion with space. It is only for the sake of studying 
the content with facility that we correlate its several objects with the points 
of space. 

398. The Intervene. 

In ordinary space the most important function of the co-ordinates of a 
pair of points is that which expresses their distance apart. We desire to 
create that function of a pair of objects which shall be homologous with 
the distance function of a pair of points in ordinary space. 

The nature of this function is to be determined solely by the attributes 
which we desire it to possess. We shall take the most fundamental pro 
perties of distance in ordinary space. We shall then re-enunciate these 
properties in generalized language, and show how they suffice to determine 
a particular function of a pair of objects. This we shall call the Intervene 
between the Two Objects. 

Let P, Q, R be three collinear points in ordinary space, Q lying between 
the other two ; then we have, of course, as a primary notion of distance, 

PQ + QR = PR. 

In general, the distance between two points is not zero, unless the points 
are coincident. An exception arises when the straight line joining the points 
passes through either of the two circular points at infinity. In this case, 
however, the distance between every pair of points on the straight line is 
zero. These statements involve the second property of distance. 

In ordinary geometry we find on every straight line one point which is 
at an infinite distance from every other point on the line. We call this the 
point at infinity. Sound geometry teaches us that this single point is 
properly to be regarded as a pair of points brought into coincidence by the 
assumptions made in Euclid s doctrine of parallelism. The existence of a 
pair of infinite points on a straight line is the third property which, by 
suitable generalization, will determine an important feature in the range. 
The fourth property of ordinary space is that which asserts that a point at 
infinity on a straight line is also at infinity on every other straight line 
passing through it. This obvious property is equivalent to a significant law 
of intervene which is vital in the theory. If we might venture to enunciate 
it in an epigrammatic fashion, we would say that there is no short cut to 
infinity. 



THE THEORY OF SCREWS TN NON-EUCLIDIAN SPACE. 435 

The fifth property of common space which we desire to generalize is 
one which is especially obscured by the conventional coincidence of the two 
points at infinity on every straight line. We prefer, therefore, to adduce 
the analogous, but more perfect, theorem relative to two plane pencils of 
homographic rays in ordinary space, which is thus stated. If the two rays 
to the circular points at infinity in one pencil have as their correspondents 
the two rays to the circular points in the other pencil, then it is easily 
shown that the angle between any two rays equals that between their two 
correspondents. 

We now write the five correlative properties which the intervene is to 
possess. They may be regarded as the axioms in the Theory of the Content. 
Other axioms will be added subsequently. 

399. First Group of Axioms of the Content. 

(I) If three objects, P, Q, R on a range be ordered in ascending para 
meter ( 400), then the intervenes PQ, QR, PR are to be so determined that 



(II) The intervene between two objects cannot be zero unless the objects 
are coincident, or unless the intervene between every pair of objects on the 
same range is also zero. 

(III) Of the objects on a range, two either distinct or coincident are at 
infinity, i.e. have each an infinite intervene with all the remainder. 

(IV) An infinite object on any range has an infinite intervene from every 
object of the content. 

(V) If the several objects on one range correspond one-to-one with the 
several objects on another, and if the two objects at infinity on one range 
have as their correspondents the two objects at infinity on the other, then 
the intervene between any two objects on the one range is equal to that 
between their correspondents on the other. 

400. Determination of the Function expressing the Intervene 
between Two Objects on a Given Range. 

Let x l , x 2 , x s , x 4 , and y 1} y z , y s , y 4 be the co-ordinates of the objects by 
which the range is determined. Then each remaining object is constituted 
by giving an appropriate value to p in the system, 



Let A. and /* be the two values of p which produce the pair of objects of 
which the intervene is required. It is plain that the intervene, whatever it 
be, must be a function of x lt x 2 , # 3 , # 4 and y j} y 2 , y s> y t , and also of X and /z. 
So far as objects on the same range are concerned, we may treat the co- 

28-2 



436 THE THEORY OF SCREWS. [400 

ordinates of the originating objects as constant, and regard the intervene 
simply as a function of X, and /*, which we shall denote by/(X, yu,). 

The form of this function will be gradually evolved, as we endow it with 
the attributes we desire it to possess. The first step will be to take a third 
object on the same range for which the parameter shall be v, where X, p, v 
are arranged in order of magnitude. Then, as we wish the intervene to 
possess the property specified in Axiom I., we have 



By the absence of /u from the right-hand side, we conclude that /j, must 
disappear identically from the left-hand side. This must be the case 
whatever X and v may be. Hence, no term in which /A enters can have 
X as a factor. It follows that f(\, p) must be simply the difference of two 
parts, one being a function of X, and the other the same function of p. 

Accordingly, we write, 



The first step in the determination of the intervene function has thus been 
taken. But the form of (f> is still quite arbitrary. 

The rank of the objects in a range may be concisely defined by the 
magnitudes of their corresponding values of p. Three objects are said to be 
ordered when the corresponding values of p are arranged in ascending or descend 
ing magnitude. 

Let P, Q, Q be three ordered objects, then it is generally impossible 
that the intervenes PQ and PQ shall be equal ; for, suppose them to be so, 
then 

PQ + QQ = PQ by Axiom I.; 

but, by hypothesis, PQ = PQ , 

and hence QQ = 0. 

But, from Axiom Tl., it follows that (Q and Q being different) this cannot be 
true, unless in the very peculiar case in which the intervene between every 
pair of objects on the range is zero. Omitting this exception, to which we 
shall subsequently return, we see that PQ and PQ cannot be equal so long 
as Q and Q are distinct. 

We hence draw the important conclusion that there is for each object P 
but a single object Q, which is at a stated intervene therefrom. 

Fixing our attention on some definite value B (what value it does not 
matter) of the intervene, we can, from each object X, have an ordered equi- 
intervene object /j, determined. Each X will define one yu,. Each p will 
correspond to one X. The values of X with the correlated values of /u, form 
two homographic systems. The relation between X and yu depends, of course, 



400] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 437 

upon the specific value of 8, but must be such that, when one of the 
quantities is given, the other shall be determined by a linear equation. It 
is therefore assumed that X and //, must be related by the equation 



where the ratios of the coefficients A, B, C, D shall depend, to some extent, 
upon 8. If X and // be a given pair of parameters belonging to objects 
at the required intervene, then 



by which the disposable coefficients in the homographic equation are reduced 
to two. 

The converse of Axiom II., though generally true, is not universally so. 
It will, of course, generally happen that when two objects coincide their 
intervene is zero. But on every range two objects can be found, each 
of which is truly to be regarded as two coincident objects of which the 
intervene is not zero. 

Let us, for instance, make X = yu, in the above equation ; then we have 



This equation has, of course, two roots, each of which points out an object 
of critical significance on the range. We shall denote these objects by 
and . Each of them consists of a pair of objects which, though actually 
coincident, have the intervene 8. The fundamental property of and is 
thus demonstrated. 

Let X be any object on the range ; then (Axiom I.) 

XO + 8 = XO; 

and as 8 is not zero, we have 

XO = infinity. 

Therefore every object on the range is at an infinite intervene from 0. A 
similar remark may be made with respect to ; and hence we learn that 
the two objects, and , are at infinity. 

We assume, in Axiom in., that there are not to be more than two objects 
on the range at infinity : these are, of course, and . We must, therefore, 
be conducted to the same two objects at infinity, whatever be the value of 
the intervene 8, from which we started.* We thus see that while the 
original coefficients A , B, C, D do undoubtedly contain 8, yet that 8 does not 
affect the equation 

A\* + (B + C) X + D = 0. 

* My attention was kindly directed to this point in a letter from Mr F. J. M Aulay. 



438 THE THE011Y OF SCEEWS. [400 

It follows that D -T- A and (B + C) -=- A must both be independent of 8. We 
may therefore make 



and thus the homographic equation becomes 



where A is the only quantity which involves 8. 

The equation can receive a much simpler form by taking the infinite 
objects as the two originating objects from which the range was determined. 
In this case the equation 



must have as roots X = and X = x, and therefore 

4=0; D = 0, 

hence the homographic equation reduces to 

B\ + C/* = ; 

since B -=- C is a function of the intervene 8, we may say, conversely, that 



We have, however, already learned that the intervene is to have the 
form 



Now we find that it can also be expressed, with perfect generality, in the 
form 



It follows that these two expressions must be equal, so that 



In this equation the particular value of B does not appear, nor is it even 
implied. The formula must therefore represent an identical result true for 
all values of X and all values of /it. 

We may, therefore, differentiate the formula with respect both to \ and 
to fji, and thus we obtain 



400] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 439 

whence we deduce 

X < (>) = /*</> (/A 

As X and p are perfectly independent, this equation can only subsist by 
assuming for < a form such that 

\<j> (\) = H, 
where H is independent of X. Whence we obtain 

</> = f, 

and, by integration, 

(X) = H log X + constant. 

The intervene is now readily determined, for 

<j> (X) - </> O) = H log X - H log p, = H log - . 

f 

We therefore obtain the following important theorem which is the well- 
known* basis of the mensuration of non-Euclidian space : 

Let #1, # 8 , #g, # 4 , and y lt y a , y s , y* be the two objects at infinity on a range, 
and let a^ + X^i, # 2 + Xy 2 , x 3 + \y 3 , x + ^y, and x l + /u# 1} x 2 + /j,y 2 , oc 3 + ^y 3 , 
i + py* be any two other objects on the range, then their intervene will be 
expressed by 

H (log X - log fi) t 

where H is a constant depending upon the adopted units of measurement. 

It will be useful to obtain the expression for the intervene in a rather 
more general manner by taking the equation in X and /A, for objects at the 
intervene 8, as 



Let X and X" be the two roots of this equation when //. is made equal to X. 
It follows that what we have just written may be expressed thus: 

X^ + X (0 - IX - IX") + fi (- 6 - ^X - IX") + X X" = 0. 

For, if X = p, this is satisfied by either X or X", while 6 disappears. 6 is, of 
course, a function of the intervene, and it is only through 6 that the inter 
vene comes into the equation. By solving for 9, we find 



* Professor George Bruce Halsted remarks in Science, N. S., Vol. x., No. 251, pages 545557, 
October 20, 1899, that "Koberto Bonola has just given in the Bolletirw di Bibliografia e Storia 
della Scienze Matematiche (1899) au exceedingly rich and valuable Bibliografia sui Fondamenti, 
della Geometria in relazione alia Geometria non-Euclidea in which he gives 353 titles." 



440 THE THEORY OF SCREWS. [400, 

The intervene itself is F (&), where F expresses some function; and, 
accordingly, 



When we substitute in this expression the value of 6 given above, we have 
an identity which is quite independent of the particular 8. We must, there 
fore, determine the functions so that this equation shall remain true for all 
values of X, and all values of p. The formulae must therefore be true when 
differentiated 

d6_ __(/*-X )OA-X") d6 = (X-X )(X"-X) 

dx t-x) 2 d^ (/i-x) 2 



whence, 



or (X - X )(X - X") < (X) = O - X )(/A - X") f (/*), 

which has the form 



Considering the complete independence of both X and /JL, this equation re 
quires that each of its members be independent alike of X and /*. We shall 
denote them by H (X X") where H is a constant, whence 

(X - X )(X - X") (j> (\) = H(\ r - X"), 






JUV 

-v x-x V 

whence, integrating and denoting the arbitrary constant by C, 

(f>(\)=H [log (X - X ) - log (X - X")] + G ; 
similarly, 



and, finally, we have for the intervene, or (X) </> (^), the expression, 



This expression discloses the intervene as the logarithm of a certain an- 
harmonic ratio. 

We may here note how a difficulty must be removed which is very 
likely to occur to one who is approaching the non-Euclidian geometry for the 



401] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 441 

first time. No doubt we find the intervene to be the logarithm of an an- 
harmonic ratio of four quantities, but these quantities are not distances 
nor are they quantities homologous with the intervene. They are simply 
numerical. The four numbers, X, jj,, X , X" are merely introduced to define 
four objects, one of them being, 



and the others are obtained by replacing X by //,, X , X", respectively. All 
we assert is, that if we choose to call the two objects defined by X and X" 
the objects at infinity, and that if we desire the intervene between the 
objects X and /j, to possess the properties that we have already specified, 
then the only function possible will be the logarithm of the anharmonic ratio 
of these four numbers. 

The word anharmonic is ordinarily applied in describing a certain 
function of four collinear points. In the more general sense, in which we 
are at this moment using the word, it does not relate to any geometrical or 
spacial relation whatever ; it is a purely arithmetical function of four abstract 
numbers. 

We may also observe that the relation between 6 and the intervene 8 
is given by the equation 

8 

P H , 1 
/i i/v \ \ 

"It* ~ x )-s- 



and the expression of the intervene as a function of 6 ; that is, the expression 
F(6} is 



401. Another process. 

We may also proceed in the following manner. Let us denote the values 
of X for the infinite objects on the range by pe ie and pe~ ie . 

If then X, fji be two parameters for two objects at an intervene , we must 
have (p. 439) 

X/i + X (e p cos 6) + fM ( e - p cos 6) + p 2 = 0. 

Solving for e, we have 

_ X/i p cos 6 (X + p,} + p z 
fji X 

The intervene 8 must be some function of e, whence 



442 THE THEORY OF SCREWS. [401- 

whence 

dF de _ dF p? - 2//, cos + p* 

* ( J de d\ ~ de ~ (fi - X) 2 

dF de _ dF A, 2 - 2X cos + p 2 

* (/ * ~~fadiM~Te ~(/a - \) 2 ~ 

(X 2 - 2X cos + p 2 ) < (A,) = (fj? - 2/4 cos + p*) < (^) = p sin suppose. 

Hence we have 

. ,. x , / p sin 6 \ 

</> (X) = tan- 1 - a - , 
\p cos \J 

and thus we get for the intervene with a suitable unit 

o, . / psin 8 \ ( p sin 6 \ 

8 = tan- ] ^ - -tan" 1 -* -^ -1. 

\p cos 6 \/ \p cos V p/ 

402. On the Infinite Objects in an Extent. 

On each range of the extent there will be two objects at infinity, by the 
aid of which the intervene between every two other objects on that range 
is to be ascertained. We are now to study the distribution of these infinite 
objects over the extent. Taking any range and one of its infinite objects, 
0, construct any other range in the same extent containing as an object. 
This second range will also have two infinite objects. Is to be one of 
them? Here we add another attribute to our, as yet, immature conception 
of the intervene. 

In Euclidian space we cannot arrive at infinity except we take an 
infinitely long journey. This is because the point at infinity on one straight 
line is also the point at infinity on any other straight line passing through 
it. Were this not the case, then a finite journey to infinity could be taken 
by travelling along the two sides of a triangle in preference to the direct 
route vid the third side. To develop the analogy between the conception of 
intervene and that of Euclidian distance, we therefore assume (in Axiom IV.) 
that an infinite object has an infinite intervene with every other object of 
the content. In consequence of this we have the general result, that 

If be an infinite object on one range, then it is an infinite object on every 
one of the ranges diverging from 0. 

The necessity for this assumption is made clear by the following con 
sideration : Suppose that were an infinite object on one range containing 
the object A, but were not an infinite object on another range OB, diverging 
also from 0; then, although the direct intervene OA is infinite, yet the 
intervenes from A to 5 and from B to would be both finite. The only 
escape is by the assumption we have just italicised. Otherwise infinity 
could be reached by two journeys, each of finite intervene. 



403 J THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 443 

Take any infinite object 0. Construct a series of ranges in the extent, 
each containing 0. Each of these ranges will have another infinite object, 
Oi, 2 , 3 , &c. The values x lt # 2 , x 3 , which define O lt 2 , 3 , &c., must fulfil 
some general condition, which we may express thus : 



Form a range through O l and 2 . There must be two infinite objects on 
this range, and of course all other objects thereon will be defined by a linear 
equation L = in x ly x 2 , x 3 . 

Every object satisfying the condition/^, x z , x 3 ) = is infinite, and there 
fore all the values of x 1} x 2 , x s common to the two equations L = and 
/(#!, a? 2 , # 3 ) = must denote infinite objects. But we have already seen that 
there are only two infinite objects on one range ; therefore there can be 
only two systems of values common to the two equations. In other words, 
f(%i, x 2 , x 3 ) must be an algebraical function of the second degree. There 
can be no infinite object except those so conditioned ; for, suppose that S 
were one, then any range through S would have two objects in common 
with /, and thus there would be three infinite objects on one range, which 
is contrary to Axiom in. 

Hence we deduce the following important result : 

All the infinite objects in an extent lie on a range of the second degree. 

We thus see that every range in an extent will have two objects in 
common with the infinite range of the second order. These are, of course, 
the two infinite objects on the range. 

403. On the Periodic Term in the Complete Expression of the 
Intervene. 

We have found for the intervene the general expression 

H (log X - log p). 
We may, however, write instead of X, 

(cos 2n7r + i sin 27r) X, 
where n is any integer ; but this equals 

e 2inir \ ; 
hence, log X = Zimr + log X ; 

and, consequently, the intervene is indeterminate to the extent of any 
number of integral multiples of 



The expression just written is the intervene between any object and 
the same object, if we proceed round the entire circumference of the range. 
We may call it, in brief, the circuit of the range. 



444 THE THEORY OF SCREWS. [403, 

The intervene between the objects X and X is 

iHir. 

Nor is this inconsistent with the fact that \ = zero denotes two coincident 
objects, as does also X= infinity. In each of these cases the coincident 
objects are at infinity, and the intervene between two objects which coalesce 
into one of the objects at infinity has an indeterminate value, and may thus, 
of course, be iH-rr, as well as anything else. 

404. Intervenes on Different Ranges in a Content. 

Let us suppose any two ranges whatever. There are an infinite number 
of objects on one range, and an infinite number on the other. The well- 
known analogies of homographic systems on rays in space lead us to inquire 
whether the several objects on the two ranges can be correlated homo- 
graphically. Each object in either system is to correspond definitely with 
a single object in the other system. 

We determine an object on a range by its appropriate X. Let the 
corresponding object on the other range be defined by X . The necessary 
conditions of homography demand that for each X there shall be one X , and 
vice versa. Compliance with this is assured when X and X are related by 
an equation of the form 

PXX + Q\ + R\ + 8 = 0. 

Let \i, X 2 , X 3 , X 4 be any four values of X, and let X/> X,/, X*/, X/ be the cor 
responding four values of X , then, by substitution in the equation just 
written, and elimination of P, Q, R, S, it follows that 

Xi Xs Xj X 4 A! Xs A-2 X4 



X^ \s Xj X 4 X 2 X 3 Xj X 4 

We now introduce the following important definition : 

By the expression, anharmonic ratio of four objects on a range, is meant 

the anharmonic ratio of the four values of the numerical parameter by which 

the objects are indicated. 

We are thus enabled to enunciate the following theorem : 

When the objects on two ranges are ordered homographically, the an 

harmonic ratio of any four objects on one range equals the anharmonic ratio 

of their four correspondents on the other. 

Three pairs of correspondents can be chosen arbitrarily, and then the 
equation last given will indicate the relation between every other X and its 
corresponding X . 

Among the different homographic systems there is one of special im 
portance. It is that in which the intervene between any two objects in 



404] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 445 

one range equals that between their correspondents in the other. But 
this homography is only possible when a critical condition is fulfilled. 

In the first place, an infinite object on one range must have, as its 
correspondent, an infinite object on the other. For if X be an infinite 
object on one range, it has an infinite intervene with every other object 
on that range (Axiom iv.) ; therefore X , the correspondent of X, must 
have an infinite intervene with every other object on the second range. 

If, then, X and Y are the infinite objects on one range, and X and Y 
the infinite objects on the other, and if A and B be two arbitrary objects on 
the first range, and A and B their correspondents on the other; then, 
using the accustomed notation for anharmonic ratio, 



But, if H be the factor (p. 440) for the second range, which H is for the first, 
we have, since the intervenes are equal, 

H log (ABXY) = H log (A B X Y ) ; 
and, since the anharmonic ratios are equal, we obtain 

H=H . 

If, then, it be possible to order two homographic systems of objects, so that 
the intervene between any two is equal to that between their correspondents, 
we must have H and H equal ; and conversely, when H and H are equal, 
then equi-intervene homography is possible. 

We have therefore assumed Axiom v. ( 399) which we have now seen 
to be equivalent to the assumption that the metric constant H is to be the 
same for every range of the content. 

Nor is there anything in Axiom V. which constitutes it a merely gratuitous 
or fantastic assumption. Its propriety will be admitted when we reduce our 
generalized conceptions to Euclidian space. It is an obvious notion that 
any two straight lines in space can have their several points so correlated 
that the distance between a pair on one line is the same as that between 
their correspondents on the other. In fact, this merely amounts to the 
statement that a straight line marked in any way can be conveyed, marks 
and all, into a different situation, or that a foot-rule will not change the 
length of its inches because it is carried about in its owner s pocket. 

In a similar, but more general manner, we desire to have it possible, on 
any two ranges, to mark out systems of corresponding objects, such that the 
intervene between each pair of objects shall be equal to that between their 
correspondents. We have shown in this Article that such an arrangement 
is possible, when, and only when, the property v. is postulated. We may 
speak of such a pair of ranges as equally graduated. 



446 THE THEORY OF SCREWS. [404- 

The conception of rigidity involves the notion that it shall be possible to 
displace a system of points such that the distance between every pair of 
points in their original position equals that between the same pair after 
the displacement. We desire to have corresponding notions in the present 
Theory, which will only be possible when we have taken such a special view 
of the nature of the intervene as is implied in Axiom v. 

405. Another Investigation of the possibility of equally Gradu 
ated Ranges. 

The importance of the subject in the last Article is so great in the 
present Theory that I here give it from a different point of view. 

Taking the infinite objects on a range as the originating objects, we have 
on the first range for the intervene between the objects X and a, 

H (log X - log a) ; 
arid for the second range for the intervene between ju, and ft, we have 



Regarding a and (3 as fixed, and X and ^ as defining a pair of correlative 
objects, we get, as the relation between X and /n for equally graduated 
ranges, 



whence the relation between X and //, is thus given : 

IT 
H 



When this is the case, there are several values of X corresponding to one 
value of p., which may be thus found : Let m be any integer ; then, in the 
usual manner, 

x f p\*( H . . H \ 

- TS cos 2w ff if + * sm *m ~fr ^ ; 

a \I3J \ H ff J 

and therefore 

fl_ 

H 



or 

fl_ 

\ H ! 2H 



406] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 447 

or 

H 1 

. 4P 



&c. &c. &c. 

We see that the correspondence between X and //. cannot be of the homo- 
graphic character, unless 

H = H . 

The necessity for this condition may be otherwise demonstrated by 
considering the subject in the following manner: 

The intervene between any two objects on one range is, of course, 
ambiguous, to the extent of any integral number of the circuits on that range. 
Let G and C be the circuits, and let 8 be an intervene between two objects 
on the second range. If we try to determine two objects, a and X, on the 
first range that shall have an intervene 8, we must also have another object 
X , such that its intervene from a is B + C . Similarly, there must be another 
object X" with the intervene S + 2C", &c. It is therefore impossible to have 
a single object at the intervene S + raC" from a, unless it happened that 

C=C , 
or that 

H = H . 

Thus, again, are we led to the conclusion that ranges cannot be equally 
graduated unless their circuits are the same. 

The circuits on every range in the content being now taken to be equal, 
we can assume for the circuit any value we please. There are great advan 
tages in so choosing our units that the circuit shall be TT ; but we have as its 
expression, 

whence we deduce 



406. On the Infinite Objects in the Content. 

Certain objects in the content are infinite, and it is proposed to determine 
the conditions imposed on x lt x^, # 3 , x 4 when they indicate one of these. If 
an object be infinite, then every range through that object will have one 
other infinite object. Let these be 1} 2 , &c. These several objects will 
conform with the condition, 



Every infinite object in the content must satisfy this equation ; and, 
conversely, every object so circumstanced is infinite. 



448 THE THEORY OF SCREWS. [406- 

Two linear equations in # 1} a? 2 , # 3 , x 4 determine a range, and the simul 
taneous solution of these equations with 



gives the infinite objects on that range but there can only be two and 
hence we have the following important theorem : 

The co-ordinates a; 1} # 2 , x 3 , x 4 of the infinite objects in a content satisfy an 
homogeneous equation of the second degree. 

We denote this equation by 



407. The Departure. 

Let the ranges x^ x 2 , 0, be formed when the parameter x 1 -r x 2 has every 
possible value, then the entire group of ranges produced in this way is called 
a star. In ordinary geometry the most important function of a pair of rays 
in a pencil is that which expresses their inclination. We have now to create, 
for our generalized conceptions, a function of two ranges in a star which 
shall be homologous with the notion of ordinary angular magnitude. 

We shall call this function the Departure. Its form is to be determined 
by the properties that we wish it to possess. In the investigation of the 
departure between two ranges, we shall follow steps parallel to those which 
determined the intervene between two objects. 

If OP, OQ, OR be three rays in an ordinary plane diverging from 0, then 



In general the angle between two rays is not zero unless the rays are coin 
cident ; but this statement ceases to be true when the vertex of the pencil is 
at infinity. In this case, however, the angle between every pair of rays in 
the pencil is zero. 

Every plane pencil has two rays (i.e. those to the circular points at 
infinity), which make an infinite angle with every other ray. 

408. Second Group of Axioms of the Content. 

We desire to construct a departure function which shall possess the 
following properties : 

(VI) If three ranges, P, Q, R, in a star, be ordered in ascending parameter, 
and if the departure between two ranges, for example, P and Q, be expressed 
by PQ, then 

PQ + QR = PR. 



410] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 449 

(VII) The departure between two ranges cannot be zero unless the ranges 
are coincident, or unless the departure between every pair of ranges in that 
star is also zero. 

(VIII) Of the ranges in a star, two (distinct or coincident) are at infinity, 
i.e. have each an infinite departure from all the remainder. 

(IX) An infinite range has an infinite departure, not only with every 
range in its star, but with every range in the extent. 

(X) If the several ranges in one star correspond one to one with the 
several ranges in another, and if the two infinite ranges in one star have as 
their correspondents the infinite ranges in the other; then the departure 
between any two ranges in one star is equal to that between the two corre 
sponding ranges in the other. 

409. The Form of the Departure Function. 

The analogy of these several axioms to those which have guided us to 
the discovery of the intervene, shows that the investigation for the function 
of departure will be conducted precisely as that of the intervene has been ; 
accordingly, we need not repeat the several steps of the investigation, but 
enunciate the general result, as follows : 

Let ac 1} x. 2 , and y 1} y z be the co-ordinates of any two ranges in a star, and 
let \i, \2, and //, 1} /i 2 be the co-ordinates of the two infinite ranges in that star. 
Then the departure between (x l , # 2 ) and (y 1} y 2 ) is 



Aa - y a X 

410. On the Arrangement of the Infinite Ranges. 

Every star in the extent will have two infinite ranges, and we have now 
to see how these several infinite ranges in the extent can be compendiously 
organized into a whole. 

To aid in this we have assumed Axiom ix., the effect of which is to 
render the following statement true. Let several objects on a range, 0, be 
the vertices of a corresponding number of stars. If be an infinite range 
in any one of the stars, then it is so in every one. 

Let a lt Oz, a s be any three ranges in an extent. Then every range in 
the same extent can be expressed by 

#!! + # 2 a 2 -f x 3 a 3 , 

where # lt x 2 , x 3 are the three co-ordinates of the range. It is required to 
determine the relation between x 1} # 2 , x 3 if this range be infinite. 

B. 29 



450 THE THEORY OF SCREWS. 

An object L will be defined by an equation of the form, where L ly L 2 , L 3 

are numerical, 

L 3 x 3 = 0, 



for any two sets of values, x lt x z , x 3 , which satisfy this equation, will deter 
mine a pair of ranges which have the required object in common. 

Let the relation between the co-ordinates of an infinite range be 

/Oi, a? 2 , s ) = 0; 

then the infinite ranges in the star, whose vertex is the object L, will be 
defined by co-ordinates obtained from the simultaneous solution of 

= 0, 



But there can only be two such ranges ; and, accordingly, the latter of these 
equations must be of the second degree. We hence deduce the following 
important result : 

The infinite ranges in an extent may be represented by the different groups 
of values of the co-ordinates x ly x z , x 3 which satisfy one homogeneous equation 
of the second degree. 

Remembering that the existence of zero intervene between every pair of 
objects on a range is a consequence of the coincidence of the two objects of 
infinite intervene on that range, we have the following result : 

The range through, two consecutive objects of infinite intervene is a range of 
zero intervene. 

And, similarly, we have the following : 

The object common to two consecutive ranges of infinite departure is the 
vertex of a star of zero departure. 

411. Relations between Departure and Intervene. 

With reference to the theory of Departure, we thus see that there is a 
system of critical ranges, and a system of critical objects in each extent. 
Every critical range has an infinite departure from every other range. Every 
critical object is the vertex of a star of which the departure between every 
pair of ranges is zero. It will be remembered that in the theory of the 
intervene we were conducted to the knowledge that every extent contained 
a system of objects and ranges, critical with regard to the intervene. Each 
critical object had an infinite intervene with every other object in the 
extent. Each critical range possessed the property that the intervene 
between any two objects thereon is zero. There are, thus, objects and 
ranges critical with repect to the intervene. There are also objects and 
ranges critical with respect to the departure. But nothing that we have 



412] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 451 

hitherto assumed will entitle us to draw any inference as to the connexion, 
much less as to the actual identity, between the critical systems related to 
intervene, and those related to departure. We have already assumed five 
properties for the intervene, and five like properties for the departure. These 
are, in fact, the axioms by which alone the functions of intervene and of 
departure could be constructed. But another axiom of quite a distinct type 
has now to be introduced. 

There are objects of infinite intervene, and objects of zero departure. 
There are ranges of infinite departure, and ranges of zero intervene. A 
range generally contains two objects of infinite intervene, and two of zero 
departure. A star generally contains two ranges of infinite departure, and 
two ranges of zero intervene. On a range of zero intervene the two objects 
of infinite intervene coalesce, and their intervene from other objects on the 
range becomes indeterminate. In a star of zero departure the two ranges 
of infinite departure coalesce, and their departure from other ranges in 
the same star becomes indeterminate. We have thus the following state 
ment : 

On a range of zero intervene, the intervene between every pair of objects 
is zero, except where one particular object is involved, in which case the 
intervene is indeterminate. 

In a star of zero departure, the departure between every pair of ranges 
is zero, except where one particular range is involved, in which case the 
departure is indeterminate. 

The new axiom to be now introduced will be formed as the others have 
been by generalization from the conceptions of ordinary geometry. In that 
geometry we have two different aspects in which the phenomenon of paral 
lelism may be presented. Two non-coincident lines are parallel when the 
ansle between them is zero, or when their intersection is at an infinite 

o 

distance. Without entering into any statement about parallel lines, we may 
simply say, that when two different straight lines are inclined at the angle 
zero, their point of intersection is at infinity. Generalizing this proposition, 
we assume the following axiom or property, which we desire that our systems 
of measurement shall possess. 

412. The Eleventh Axiom of the Content. 

This axiom, which is the first to bring together the notions of intervene 
and of departure, is thus stated : 

(xi) If two ranges in the same extent have zero departure, their common 
object will be at infinity, and conversely. 

The vertex of every star of zero departure will thus be at infinity, and 
hence we deduce the important result that all the objects of infinite inter- 

292 



452 THE THEORY OF SCREWS. [412, 

vene are also objects of zero departure, and conversely. Thus we see that 
the two systems of critical objects in the extent coalesce into a single 
system in consequence of the assumption in Axiom xi. 

Each consecutive pair of critical objects determine a range, which is a 
range of infinite departure as well as of zero intervene. 

The expression infinite objects will then denote objects which possess the 
double property of having, in general, an infinite intervene from other objects 
in the extent, and of being also the vertices of stars of zero departure. 

The expression infinite ranges will denote ranges which possess the double 
property of having, in general, an infinite departure with all other ranges, 
and which consist of objects, the intervene between any pair of which is, in 
general, zero. 

There is still one more point to be decided. The measurement of depar 
ture, like that of intervene, is expressed by the product of a numerical 
factor with the logarithm of an anharmonic ratio. This factor is H for the 
intervene. Let us call it H for the departure. What is to be the relation 
between H and H l Here the analogy of geometry is illusory; for, owing 
to the coincidence between the points of infinity on a straight line, H has to 
be made infinite in ordinary geometry, while H must be finite. But in the 
present more general theory H is finite, and we have found much convenience 

*t 

derived from making it equal to -= , for then the entire circuit of any range 

z 

ft 

is TT. We now stipulate that H is also to be ^ . The circuit of a star 

it 

will then be TT also. 

With this assumption the theory of the metrics of an extent admits of a 
remarkable development. 

Let x, y, z be any three objects. Let a, b. c denote the intervenes 
between y and z, z and x, x and y, respectively. Let the departure between 
the ranges from x to y and x to z be denoted by A, from y to z and y to # be 
denoted by B, and from z to x and z to y be denoted by C. Then, 

sin A _ sin B _ sin G 
sin a sin b sin c 

cos a = cos b cos c 4- sin b sin c cos A, 
cos b = cos c cos a + sin c sin a cos B, 
cos c = cos a cos b + sin a sin b cos C. 

Thus the formulae of spherical trigonometry are generally applicable through 
out the extent*. 

* I learned this astonishing theorem from Professor Heath s very interesting paper, Phil. 
Trans. Part n. 1884. 



413] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 453 

413. Representation of Objects by Points in Space. 

The several objects in a content are each completely specified when the 
four numbers, x lt x 2 , x 3 , # 4 , corresponding to each are known. It is only the 
ratios of these numbers that are significant. We may hence take them to 
be the four quadriplanar co-ordinates of a point in space. We are thus led 
to the construction of a system of one-to-one correspondence between the 
several points of an Euclidian space, and the several objects of a content. 
The following propositions are evident : 

One object in a content has for its correspondent one point in space, and 
one point in space corresponds to one object in the content. 

The several objects on a range correspond one to one with the several points 
on a straight line. 

The several objects in an extent correspond one to one with the several 
points in a plane. 

Since the objects at infinity are obtained by taking values of x l} x z , x 3 , x t , 
which satisfy a quadric equation, we find that 

The several objects at infinity in the content correspond with the several 
points of a quadric surface. 

This surface we shall call the infinite quadric. 

The following theorem in quadriplanar co-ordinates is the foundation of 
the metrics of the objects in the content by the points in space. 

If # 1; #2, s, \ and y 1} y 2 , y*, y* be the quadriplanar co-ordinates of two 
points P and Q respectively, and if 1} 2 , 3 , 4 be any other four points on 
the ray PQ whose co-ordinates are respectively 



i, #2+^42/2, %3 + \y 3 , #4 + ^42/4, 

then, we have the following identity 



Remembering the definition of the anharmonic ratio of four objects on a 
range ( 404), we obtain the following theorem : 

The anharmonic ratio of four objects on a range equals the anharmonic 
ratio of their four corresponding points on a straight line. 



454 



THE THEORY OF SCREWS. [413- 



We hence deduce the following important result, that 

The intervene between any two objects is proportional to the logarithm of 
the anharmonic ratio in which the straight line joining the corresponding 
points is divided by the infinite quadric. 

We similarly find that 

The departure between any two ranges in the same extent is proportional 
to the logarithms of the anharmonic ratio of the pencil formed by their two 
corresponding straight lines, and the two tangents in the same plane from 
their intersection to the infinite quadric. 

414. Poles and Polars. 

The point # 1} x z , x 3 , x 4 has for its polar, with regard to the infinite quadric, 
the plane, 

dU dU dU dU A 

X 1 3 h X 2 -j h X. A -j h # 4 -7 = 0. 

dxi dx 2 dx 3 dxi 

Thus we see that an object corresponding to the point will have a polar 
extent corresponding to the polar of that point with regard to the infinite 
quadric. The following property of poles and polars follows almost imme 
diately. 

7T 

The intervene from an object to any object in its polar extent is equal to -^ . 

We have hitherto spoken of the departure between a pair of ranges 
which have a common object: we now introduce the notion of the departure 
between a pair of extents by the following definition : 

The departure between a pair of extents is equal to the intervene between 
their poles. 

415. On the Homographic Transformation of the Content. 

In our further study of the theory of the content we shall employ, 
instead of the objects themselves, their corresponding points in ordinary 
space. All the phenomena of the content can be completely investigated 
in this way. Objects, ranges, and extents, we are to replace by points, 
straight lines, and planes. Intervenes are to be measured, not, indeed, as 
distances, but as logarithms of certain anharmonic ratios obtained by ordinary 
distance measurement. Departures are to be measured, not, indeed, as 
angles, but as logarithms of anharmonic ratios of certain pencils obtained 
by ordinary angular measurement. 

I now suppose the several objects of a content to be ordered in two 
homographic systems, A and B. Each object, X, in the content, regarded 
as belonging to the system A, will have another object, Y, corresponding 
thereto in the system B. The correspondence is to be simply of the one-to- 



416J THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 455 

one type. Each object in one system has one correspondent in the other 
system. But if X be regarded as belonging to the system B, its correspondent 
in A will not be Y, but some other object, Y . 

To investigate this correspondence we shall represent the objects by 
their correlated points in space. We take x l} x 2 , x 3 , x 4 as the co-ordinates 
of a point corresponding to as, and y l} y 2 , y 3 , y 4 as the co-ordinates of the 
point corresponding to y. We are then to have an unique correspondence 
between as and y, and we proceed to study the conditions necessary if this 
be complied with. 

416. Deduction of the Equations of Transformation. 

All the points in a plane L, taken as x points, must have as their 
correspondents the points also of a plane ; for, suppose that the corre 
spondents formed a surface of the nth degree, then three planes will have 
three surfaces of the nth degree as their correspondents, and all their n 3 
intersections regarded as points in the second system will have but the 
single intersection of the three planes as their correspondent in the first 
system. But unless n = 1 this does not accord with the assumption that the 
correspondence is to be universally of the one-to-one type. Hence we see 
that to a plane of the first system must correspond a plane of the second 
system, and vice versa. 

Let the plane in the second system be 

AM + A. 2 y, + A 3 y, + A 4 y t = 0. 

If we seek its corresponding plane in the first system, we must substitute 
for y 1} 7/2, 2/s> 2/4 the corresponding functions of x ly x. 2> x 3 , # 4 . Now, unless 
these are homogeneous linear expressions, we shall not find that this remains 
a plane. Hence we see that the relations between X 1) x 2 ,x 3 ,x 4 and y\,y<i>y*,yi 
must be of the following type where (11), (12), &c., are constants : 

TA = (11K + (12) * 2 + (13) * 3 + (14) * 4 , 

7/ 2 = (21) x, + (22) x z + (23) x s + (24) a> 4 , 

y , = (31 K + (32) x, + (33)^3 + (34) # 4 , 

/4 = (41 X + (42)^ 2 + (43)^ 3 + (44) x 4 . 

Such are the equations expressing the general homographic transformation 
of the objects of a content. From the general theory, however, we now 
proceed to specialize one particular kind of homographic transformation. 
It is suggested by the notion of a displacement in ordinary space. The 
displacement of a rigid system is only equivalent to a homographic trans 
formation of all its points, conducted under the condition that the distance 
between every pair of points shall remain unaltered (see p. 2). In our extended 
conceptions we now study the possible homographic transformations of a 



456 THE THEORY OF SCREWS. [416, 

content, conducted subject to the condition that the intervene between 
every pair of objects shall equal that between their correspondents. 

417. On the Character of a Homographic Transformation 
which Conserves Intervene. 

In all investigations of this nature the behaviour of the infinite objects 
is especially instructive. In the present case it is easily shown that every 
object infinite before the transformation must be infinite afterwards 
when moved to . For, let X be any object which is not infinite before the 
transformation, nor afterwards, when it becomes X . Then, by hypothesis, 
the intervene OX is equal to O X ; but OX is infinite, therefore O X must 
be also infinite, so that either or X is infinite ; but, by hypothesis, X is 
finite, therefore must be infinite, so that in a homographic transformation 
which conserves intervene, each object infinite before the transformation remains 
infinite afterwards. 

It follows that in the space representation each point, representing 
an infinite object, and therefore lying on the infinite quadric U=Q must, 
after transformation, be moved to a position which will also lie on the 
infinite quadric. Hence we obtain the following important result : 

In the space representation of a homographic transformation which con 
serves intervene, the infinite quadric U=Q is merely displaced on itself. 

A homographic transformation of the points in space will not, in general, 
permit any quadric to remain unchanged. A certain specialization of the 
constants will be necessary. They must, in fact, satisfy a single condition, 
for which we shall presently find the expression. 

Let cc lt oc 2 , x 3 , # 4 be the quadriplanar co-ordinates of a point, and let us 
transform these to a new tetrahedron of which the vertices shall have as 
their co-ordinates with respect to the original tetrahedron 

1 . rr /y> 

"I ) *2 ) X 3 > " 4 > 

v" <r" r" <r" 

*Tl i a 2 ^3 > "^l > 



If then X l} X 2 , X 3 , X 4 be the four co-ordinates of the point referred to 
the new tetrahedron 



+ Xi X* + x^"X s + 



417] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 457 



For these equations must be linear and if X z , X 3 , X^ are all zero then 
#i> 2, 3, #4 become a;/, # 2 , x 3 , x as they ought to do, and similarly for the 
others, whence we get 

X,= 



vCo OC*> 



OCn Otif? vu- 



(KA oc* 



We may write this result thus 



Let us now suppose that the vertices of this new tetrahedron are the 
double points of a homography defined by the equations 

2h = (11) x, + (12) x, + (13) x, + (14) x 4t 
y, = (21) x, + (22) x 2 + (23) 8 + (24) a? 4 , 
y 3 = (31) ^ + (32) ^ 2 + (33) x t + (34) # 4 , 
y 4 = (41) ^ + (42) a; 2 + (43) x a + (44) ar 4 . 

We have to solve the biquadratic 

(11) -p (12) (13) (14) =0. 

(21) (22) -p (23) (24) 

(31) (32) (33) -p (34) 

(41) (42) (43) (44) -p 

Let the roots be p l} p^, p 3 , p t . Then we have 

Pl x{ = (11) / + (12) / + (13) a?, + (14) ar 4 , 
p a a7 2 = (21) a?/ + (22) # 2 + (23) x, + (24) a?/, 
/9^ 3 = (31 ) a?/ + (32) a? s + (33) ar, + (34) ar/, 
P!^ = (41) a;/ + (42) a?/ + (43) <c, + (44) a-/, 

with similar equations for a;/ , a;/", a^"", a;/ , &c. 



458 



THE THEORY OF SCREWS. 



[417 



We thus get 

Hpipipsp+Xi = 



or 



H Pl X^ 



11 12 13 14 

21 22 23 24 

31 32 33 34 

41 42 43 44 



2/1 y* 2/3 2/4 

X-i Xn Xv XA 



But these determinants are the co-ordinates of y referred to the new 
tetrahedron, and omitting needless factors 



We thus obtain the following theorem. 

Let x ly x 2 , x 3 , #4 be the co-ordinates of a point with respect to any arbitrary 
tetrahedron of reference. 

Let T/J, y. 2 , t/ 3 , y be the co-ordinates of the corresponding point in a 
homographic system defined by the equations 

y l = (11) x l + (12) x z + (13) x 3 + (14) x, 
y z = (21) as, + (22) x, + (23) x, + (24) * 4 , 
y z = (31) x, + (32) x 2 + (33) x, + (34) x,, 
7/ 4 = (41) x, + (42) x, + (43) x, + (44) x.. 

If we transform the tetrahedron of reference to the four double points of 
the homography, and if X 1} X 2 , X 3 , X^ be the co-ordinates of any point 
with regard to this new tetrahedron then the co-ordinates of its homographic 
correspondent are 



=0. 



where p 1} p 2 , p 3 , p 4 are the four roots of the equation, 

(11) -p (12) (13) (14) 

(21) (22) -p (23) (24) 

(31) (32) (33) -p (34) 

(41) (42) (43) (44) -p 



417] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 459 

In general there are four, but only four double points, i.e. points which 
remain unaltered by the transformation. If however two of the roots of 
the biquadratic equation be equal, then every point on the ray connecting 
the two corresponding double points possesses the property of a double 
point. 

For if p 1 = p 2 , then 

YI : Y 2 :: X 1 : X 2 , 

and hence the point whose co-ordinates are 

X,, Z a , 0, 0, 

being transformed into 

F 1; F 2 , 0, 
remains unchanged. 

Let us now suppose that a certain quadric surface is to remain unaltered 
by the homographic transformation. 

At this point it seems necessary to choose the particular character of the 
quadric surface in the further developments to which we now proceed. The 
theory of any non-Euclidian geometry will of course depend on whether the 
surface adopted as the infinite be an ellipsoid or a double sheeted hyperboloid 
with no real generators or a single sheeted hyperboloid with real generators. 
We shall suppose the infinite, in the present theory, to be a single sheeted 
hyperboloid. 

The homographic transformation which we shall consider will transform 
any generator of the surface into another generator of the same system, for 
if it transformed the generator into one of the other system, then the 
two rays would intersect, which is a special case that shall not be here 
further considered. 

Let three rays R 1} R 2 , R s be generators of the first system on the 
hyperboloid. After the transformation these rays will be transferred to 
three other positions jR/, R 2 , R 3 belonging to the same system. 

Let S 1} 83 be two rays of the second system. Then the intersection of 
R lt RI, R 2 , R 2 &c., with Si give two systems of homographic points. The 
two double points of these systems on Si give two points through which two 
rays of the first system must pass both before and after the transformation. 
Two similar points can also be found on S 2 . These two pairs of Double 
points on Si and $ 2 will fix a pair of generators of the first system which are 
unaltered by the transformation. 



460 THE THEORY OF SCREWS. [417, 

In like manner we find two rays of the second system which are unaltered. 
The four intersections of these rays must be the four double points of the 
system. 

We can also prove in another way that in the homographic transforma 
tion which preserves intervene, the four double points must, in general, lie 
on the fundamental quadric. 

For, suppose that one of the double points P was not on the quadric. 
Draw the tangent cone from P. The conic of contact will remain unaltered 
by the transformation. Therefore two points 1 and 0. 2 on that conic will 
be unaltered (p. 2). So will R the intersection of the tangents to the conic 
at Oj and 2 . The four double points will therefore be P, R, O l and 2 . 

But PR cuts the quadric in two other points which cannot change. 
Hence PR will consist entirely of double points, and therefore the discri 
minant of the equation in p would have to vanish, which does not generally 
happen. 

Of course, even in this case, there are still four double points on the 
quadric, i.e. O ly 0. 2 and the two points in which PR cuts the quadric. 

We may therefore generally assume that two pairs of opposite edges of 
the tetrahedron of double points are generators of the fundamental quadric, 
the latter must accordingly have for its equation 



with the essential condition 



Every point on any quadric of this family will remain upon that quadric 
notwithstanding the transformation. 

Nor need we feel surprised, when in the attempt to arrange a homographic 
transformation which shall leave a single quadric unaltered, it appeared 
that if this was accomplished, then each member of a family of quadrics 
would be in the same predicament. Here again the resort to ordinary 
geometry makes this clear. 

In the displacement of a rigid system in ordinary space one ray remains 
unchanged, and so does every circular cylinder of which this ray is the 
axis. Thus we see that there is a whole family of cylinders which remain 
unchanged ; and if U be one of these cylinders, and V another, then all the 
cylinders of the type U+XV are unaltered, the plane at infinity being of 
course merely an extreme member of the series. More generally these 
cylinders may be regarded as a special case of a system of cones with a 
common vertex ; and more generally still we may say that a family of 
quadrics remains unchanged. 



418] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 461 

Reverting, then, to a space of which the several points correspond to the 
objects of a content, we find, that for every homographic transformation 
which corresponds to a displacement in ordinary geometry a singly infinite 
family of quadrics is to remain unchanged, and the infinite quadric itself is 
to form one member of this family. 

Let us now suppose a range in the content submitted to this description 
of homographic transformation. Let P, Q be two objects on the range, and 
let X, Y be the two infinite objects thereon. This range will be transformed 
to a new position, and the objects will now be P , Q , X , Y . Since infinite 
objects must remain infinite, it follows that X and Y must be infinite, as 
well as X and Y. Also, since homographic transformation does not alter 
anharmonic ratio, we have 

(PQXY) = (P Q X Y }; 

whence, by Axiom v., we see that the intervene from P to Q equals the 
intervene from P to Q ; in other words, that all intervenes remain unchanged 
by this homographic transformation. 

Every homographic transformation which possesses these properties must 
satisfy a special condition in the coefficients. This may be found from 
the determinantal equation for p (p. 458), for then the following symmetric 
function of the four roots p lt p. 2 , p :i , p 4 must vanish : 

(Pi Pz ~ Pap*) (pip 3 - p 2 pi) (pip* - p 2 p s ). 

418. The Geometrical Meaning of this Symmetric Function. 

We may write the family of quadrics thus : 

JVXs+ : xr t x-.a, 

All these quadrics have two common generators of each kind : 
, = 0, ^ 3 = fZ 1 = 0, X, = 

and and 



For the rays ^ = 0, X a = Q, and X 1 = 0, Z 4 = 0, are both contained in the 
plane X 1; and therefore intersect, and, accordingly, belong to the opposed 
system of generators. 

The geometrical meaning of the equation 

PiPz Psp* 

can be also shown. 

The tetrahedron formed by the intersection of the two pairs of generators 
just referred to remains unaltered by the transformation. Any point on the 
edge, X l = 0, X 3 = 0, of which the co-ordinates are 

0, *, 0, Z 4 , 



462 THE THEORY OF SCREWS. 

will be transformed to 

0, p 2 X z , 0, p 4 X 4 . 

The question may be illustrated by Figure 45. 



[418 




Fig. 45. 

Let 1, 2, 3, 4 be the four corners of the tetrahedron. Let the transfor 
mation convey P to P and Q to Q . As P varies along the ray, so will P 
vary, and the two will describe homographic systems, of which 2 and 4 are 
the double points. In a similar way, Q and Q will trace out homographic 
systems on the ray 1 3. We shall write the points on 2 4, in the order, 

2, 4, P, P . 

Through 2, the generator of the surface 2 3 can be drawn (1 2 is not a 
generator), and through 4 the generator 4 1 can be drawn (4 3 is not a 
generator) ; thus we have, for the corresponding order on 1 3. 



3, 1, Q, Q . 



Points. 



Points. Co-ordinates. 

20100 

40001 
P X 2 X 
P o P*X 2 p,X 



The anharmonic ratio of the first set is that of 0, 
,, second 



Co-ordinates. 



3 








1 





1 


i 











Q 


I/ 





ar; 





Q i 


piXS 





p 3 Z 3 






Z/ V 

3 ^3-^-3 

, 0, y-,, -yvJ 

Aj ^ A! 



but since 

then the anharmonic ratios are equal. 



418] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 463 

The theorem can otherwise be shown by drawing Figure 46. 




Fig. 46. 



1 4 and 3 2 are to be generators of the infinite quadric. This will show 
that 4 (and not 2) is the correspondent to 1, and that 2 (and not 4) is the 
correspondent to 3, and thus the statement of anharmonic equality, 

(1 QQ 3) = (4 PP 2), 
becomes perfectly definite. 

1 2 and 3 4 are, of course, not generators; they are two conjugate polars 
of the infinite quadric. 

We can now see the reason of the anharmonic equality. Let PQ be a 
generator of the infinite quadric, as is clearly possible, for 1, 3 and 2, 4 are 
both generators of the opposite system. Then, since a generator of the 
infinite quadric must remain thereon after the displacement, it will follow 
that P Q , to which PQ is displaced, must also be a generator ; and thus 
we have four generators, 4 1, PQ, P Q , 2 3, on a hyperboloid of one system 
intersecting the two generators of another, and by the well-known property 
of the surface, 



We also see why the infinite quadric is only one of a family which remains 
unaltered. For, if PQ be a generator of any quadric through the tetrahedron, 
1, 2, 3, 4; then, since P and Q are conveyed to P and Q , and since the 
anharmonic equality holds, it follows that P Q will also be a generator of 
the quadric, i.e. a generator of the quadric will remain thereon after the 
displacement. 

It is a remarkable fact that, when the linear transformation is given, the 
infinite quadric is not definitely settled. We have seen how, in the first 
place, the linear transformation must fulfil a fundamental condition; but 
when that condition is obeyed, then a whole family of quadrics present 
themselves, any one of which is equally eligible for the infinite. 



464 THE THEORY OF SCREWS. [419, 

419. On the Intervene through which each Object is Conveyed. 

Given an object, X 1} X 2 , X s , X 4 , find the intervene through which it is 
conveyed by the transformation, when 



is the infinite quadric. 

In this equation substitute for X l} X l + 6Y 1> &c.; and, remembering that 
Y l = p^X-L, &c., we have, 



(X, + pjx,) (X, 

or 6- (p 1 p 2 X l X 2 + \p 3 p 4 X 3 X 4 ) 

+ 6 (p^X^X^ + p 2 X l X 2 + Xp 3 X 3 X 4 + \p 4 X s X 4 ) 
+ X,X 2 + \X 3 X 4 = 0. 

We simplify this by introducing 

Pi 02 = PS Pl, 

and writing \X 3 X t -=- XX Z = (f>, whence the equation becomes 

ffifrpi (1 + 0) + 6 [ Pl + p. 2 + </> ( Pa + p,)] + (1 + </>) = ; 
hence if 8 be the intervene, we have, 
cos 8 = 



2 V/hp. 1 + < 

or, if we restore its value to <f>, 

cos g _ 1 ^1^2 (pi + p 2 H_(p 3 + p 4 

V^L/o \j . . . ^^ ^^ 

H Pt + P2 = Pi + P*, 

then cosS = ^il: 



i.e. all objects are translated through equal intervenes. This is the case 
which we shall subsequently consider under the title of the vector, as this 
remarkable conception of Clifford s is called. In this case, as 

and also, p l p 2 = p 3 p^ } 

we must have PI = p s , and p 2 = p 4 , 

or Pi = p t , and p a = p a . 

In either case the equation for p will become a perfect square. 



420] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 465 

In general let X 1 = 0, 

then cos 8 = p3 tjgi 



whence we find that all objects in the extent Z a are displaced through equal 
intervenes. This intervene can be readily determined for 



whence 



or 



8 = 



The intervene through which every object on X 2 is conveyed has the 
same value. 

We could have also proved otherwise that objects on X 1 and X^ are all 
displaced through equal intervenes, for the locus of objects so displaced is 
a quadric of the form 

XtXt+XX^Tj-t), 

and, of course, for a special value of the distance this quadric becomes 
simply 

^JT.-O. 

If X l = 0, and X 3 = 0, then cos 8 becomes indeterminate ; but this is as 
it should be, because all objects on X l and X 3 are at infinity. 

420. The Orthogonal Transformation*. 

The formulae 



y, = (21) Xl + (22) x, + (23) x 3 + (24) x ti 

7/3 = (31) x, + (32) # 2 + (33) x, + (34) a? 4 , 

2/4 = (41 ) x, + (42) x z + (43) ^ + (44) x 4 , 

denote the general type of transformation. The transformation is said to be 
orthogonal if when x lt &c., are solved in terms of y lt &c. we obtain as follows: _ 

X, = (11) yi + (21) y, + (31) y, + (41) y 4 , 

^ 2 = (12) y, + (22) y a + (32) y, + (42) y 4> 

^3 = (13) yi + (23) y a + (33) y s + (43) y 4 , 

** = (14) y x + (24) y a + (34) y, + (44) y 4 . 
This is employed in Professor Heath s memoir, cited on p. 452. 
B " 30 



466 



THE THEORY OF SCREWS. 



[420- 



From the first formulae the equation for p is, as before, 417 



(11) -p (12) (13) (14) 

(21) (22) -p (23) (24) 

(31) (32) (33) -p (34) 

(41) (42) (43) (44) -p 

From the second, the equation for p must be 



=0. 



<">-$ 

(12) 
(13) 
(14) 


(21) 
(22) -1 
(23) 

(24) 


(31) 
(32) 
(33) -1 
(34) 


(41) 
(42) 
(43) 
(44)- 


1 

P 



= 0; 



but we may interchange rows and columns in a determinant so that the last 
may be written, 

= 0; 



(11)- 


(12) 


(13) 


(14) 




p 










(21) 


(22) -1 


(23) 


(24) 






P 








(31) 


(32) 


(33) - 1 


(34) 








P 






(41) 


(42) 


(43) 


(44)- 


1 

P 



whence we see that the equation for p must be unaltered, if for p we sub 
stitute - . It must therefore be a reciprocal equation of the type 

p 4 + 4Ap* + GBp 2 + 4<Ap +1=0, 
and the roots are of the form 

, 1 ]_. 
and as 



this transformation fulfils the fundamental condition ( 417). 

421. Quadrics unaltered by the Orthogonal Transformation. 

The special facilities of the orthogonal transformation in the present 
subject arise from the circumstance that it is the nature of this transforma 
tion to leave unaltered a certain family of quadrics. This is as we have 



422] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 467 

seen the necessary characteristic of the homographic transformation which 
preserves intervene. The infinite quadric which the transformation fails 
to derange can be written at once, for we have 

x* + x? + # 3 2 + # 4 2 = yi 2 + / 2 2 + y? + y? = 0. 

It is also easily seen that the expression 



is unchanged by the orthogonal transformation. We thus have the following 
quadric, which remains unaltered : 



+ (24K] 
^ 
or, writing it otherwise, 



If this be denoted by U, and x? + x + x + # 4 2 by H, then, more generally, 
U AH is unaltered by the transformation. 

We now investigate the intervene 6, through which every object on 

V-MI = Q 

is conveyed by the transformation. 

If we substitute x 1 + \y 1 &c. for x l &c. in the infinite quadric we have 



and, accordingly, the intervene 6, through which an object is conveyed by 
the orthogonal transformation is defined by the equation 

cos ^ = n ; 

hence the locus of objects moved through the intervene is simply 

u - n cos e = o. 

422. Proof that U and H have Four Common Generators. 

The equation in p has four roots of the type 



These correspond to the vertices of the tetrahedron (fig. 47). Symmetry 
shows that the conjugate polars as distinguished from the generators will be 
the ray joining the vertices corresponding to 

and p , * 

302 



468 THE THEORY OF SCREWS. [422- 

and that joining 



p 




Fig. 47. 

Let a 1} 2 , 3 , 4 and /3 1} /9 2 , /3 3) /3 4 be the co-ordinates of the corners at p 
and p". If we substitute a : + X,^, a 2 4- X/3 2 . . . &c., for x 1} x 2 &c. in fl = 0, 

2\ (a, J3, + a 2 /3 2 + a 3 /3 3 + 4 /3 4 ) = 0. 
Let us make the same substitution in U, we have, in general, 



= (11) (a, + X&) + (12) (a, + \/3 2 ) + (13) (a, + X&) + (14) (a 4 + X/3 4 ) 
= p otj + Xp"/3i ; 
whence, remembering that 



+ \p"/3 2 ) + &c., 
and as a and /S are both on O, we have, 

U=\(p+ p") ( ai /3 x + 2 /9 2 + a 3 yS 3 + a 4/ S 4 ) ; 

but since the line joining p and p" is a generator of O, the last factor must 
vanish, and the line is therefore also a generator of U. 

It is thus proved that U has four generators in common with fl. 
423. Verification of the Invariance of Intervene. 

As an exercise in the use of the orthogonal system of co-ordinates, we 
may note the following proposition : 

Let #!, x. 2 , x 3 , # 4 , and #/, a? 2 , x 3 , #/, be two objects which are conveyed by 
the transformation to y lt y 2 , y a , 7/ 4 , and y/, y a , y 3 , y 4 , respectively, it is 
desired to show that the intervene between the two original points is equal 
to that between the transformed. The expressions for the cosine of the 
intervene are 



(x* + x? + x? + xff (x^ + x? + x^ + 



424] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 469 

and the similar one with y and y , instead of x and x . The denominators 
are clearly equal, and we have only to notice that 



as an immediate consequence of the formula? connecting the orthogonal 
transformation. 

424. Application of the Theory of Emanants. 

We can demonstrate the same proposition in another manner by revert 
ing to the general case. 

Let U=0 be a function of x 1} x 2 , x 3 , # 4 . Let #/, x 2 , x 3 , x 4 be a system 
of variables cogredient with x lt x. 2 , x 3 , # 4 , and let us substitute in U the ex 
pressions x l + kxi, # 2 + kx 2 , &c., for x lt x 2 . The value of U then becomes 



where 



. _ , d , d , d , d 

LA = Xi j -J- X.2 i ~ ~\~ Xn ~^j -p X z 



If U be changed into V, a function of y, by the formulae of transformation, 
we have, of course, 

U-Yt 

but since y l is a linear function of x l} &c., i.e. 

y 1 = (11) x, + (12) x 2 + (13) x 3 + (14) x., 

it follows that if we change x^ into x 1 + kx 1 , &c., we simply change y^ into 
y l + ky l . Hence we deduce, that if U be transformed by writing x 1 + kx 1 , 
&c., for x, then V will be similarly transformed by writing y^ + %/ for y, 
and, of course, as the original U and V were equal, so will the transformed 
U and V be equal. It further follows that as k is arbitrary, the several 
coefficients will also be equal, and thus we have 



=y l , 
l dy, 

Hence the intervene between two objects before displacement remains 
unaltered by that operation ; for 

,dU 
* dx +&C 

rx _ (Mil 



and by what we have just proved, this expression will remain unaltered if 
y be interchanged with x. 



470 THE THEORY OF SCREWS. 

425. The Vector in Orthogonal Co-ordinates. 

Since, in general, 

cos 6 = pr , 

12> 

we have for the vector ( 419) the following conditions : 

(11) = (22) = (33) = (44), 
and also, 



[425 



and the similar equations. In fact, U can only differ from fl by a constant 
factor. 

The orthogonal equations require the following conditions 
+ (11) . (12) - (12) . (11) + (13) . (23) + (14) . (24) = 0, 
+ (11) . (13) - (12) . (23) - (13) . (11) + (14) . (34) = 0, 
+ (11) . (14) - (12) . (24) - (13) . (34) - (14) . (11) = 0, 
+ (12) . (13) + (11) . (23) - (23) . (11) + (24) . (34) = 0, 
+ (12) . (14) + (11) . (24) - (23) . (34) - (24) . (11) = 0, 
+ (13) . (14) + (23) . (24) + (11) . (34) - (11) . (34) = 0, 

h(14) 2 =l, 
h(24) 2 =l, 
+ (13) 2 + (23) 2 + (11)- + (34) 2 = 1, 
+ (14) 2 + (24) 2 + (34) 2 + (II) 2 = 1. 

We now introduce the notation : 

(!!) = ; (12) = 0; (13) = 7; (14) = 8, 
and the equations give us 

+ 7 (23)+ B (24) = (i), 

-(23) + a (34) = 

-(24)- 7 (34) = 

+ 7 +(24)(34) = 






+ a 2 + p 2 
+ /3 2 + a 2 
+ 7 2 + (23) 



-(23)(34) = 
+(23)(24) = 
+ 7 2 + 8- = 1 
+ (23) 2 + (24) 2 =l 
+ a 2 + 



(34) 2 =l 
a 2 -l) 



(ii), 
, (iii), 

. (iv), 

(vi), 
(vii). 



425] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 471 



From (iv), 

From (v), 

by multiplication, 

but, from (vi), 
whence, we deduce, 



= -(24) (34). 
= + (23) (34); 

= -(23)(24)(34) 2 ; 
7 S = -(23)(24); 

ft 2 - (34) 2 . 



The significance of the double sign in the value of ft will be afterwards 
apparent ; for the present we take 



From (ii) B = + (23), 

From (iii) 7 = - (24), 

while the group (vii) will be satisfied if 



The scheme of orthogonal transformation for the Right Vector (for so we 
designate the case of ft = + (34),) is as follows : 

+ a + ft +7 + B 

-/3 + a. + B - 7 

-7 - B + a +/3 

- B +7 -ft + a 
If we append the condition 

then we have completely defined the Right Vector. 
We now take the other alternative, 

= -(34); 

then, from (ii), B = (23), 

then, from (iii), 7 = + (24). 

We thus have for the Left Vector, the form, 

+ a + ft +7 + B 
ft + a. 8 +7 
-7 +8 + a -ft 

- B -7 +/3 + a 



472 



THE THEORY OF SCREWS. 



[425- 



with, as before, the condition, 

a 2 + /3 2 + r + S 2 = 1- 

If 6 be the intervene through which the vector displaces an object then 
it is easily shown that cos 6 = c/. 

426. Parallel Vectors. 

The several objects of a content are displaced by the same vector along 
ranges which are said to be parallel. 

Taking the space representation, 413, Clifford showed that all right 
vectors, which are parallel, intersect two generators of one system on the 
infinite quadric, while all left vectors, which are parallel, intersect two 
generators of the other system. 

A generator intersected by two rays from a right vector may be defined by 
the points whose coordinates are 

+ < -/3, -7, -B, 

+ & +a , -8, +7, 

while a generator intersected by two rays from a left vector will be 
defined by 

+ <*o, - fio, ~7o> -&o, 



To prove the theorem, it is only necessary to show that these four points 
are coplanar, for then the two generators intersect, i.e. are of opposite 
systems. We have, then, only to show that the following determinant 
vanishes : 

7 8 



a 



a 



70 



This will be most readily shown by squaring, for with an obvious notation 
it then reduces to the simple form 







O a ] 



[ /?] 







whence we see that the original determinant is simply 

[ ] [/8o ] 



aa fl - Q - 



427] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 473 

which expanded, becomes 

(a + ftj3 + 770 + 88 V 
(a po py 760 + 070 

= a 2 2 + # 2 /3 2 -I- a 2 /3 2 + /2 /3 2 - 7 2 7 2 - S 2 S 2 - &y<? - 7 2 S 2 
= a 2 (a 2 + /3 2 ) + ^ (a /2 + /3 2 ) - 7 2 (7 2 + S 2 ) - 3 2 (7 2 + S 2 ) 
= (a 2 -I- /3 2 ) (a,, 2 + /3 2 ) - (7 2 + S 2 ) (7o 2 + 8 2 ) 5 

but, a 2 +/3 2 + 7 2 + 8 2 = 0; 

whence this expression is 

( a /2 + /3 2 )(a 2 +/3 2 + 7 2 +S 2 ) = 0. 

On the supposition that the vectors were homonymous, i.e. both right or 
both left, the corresponding determinant would have been 

a yS 7 8 

n i ^ 

p a 6 7 

o - @o 7o &o 

/3o a/ -^o 7o 
Squaring, we get, as before, 



but now, 



whence the determinant reduces to 

[aV] 2 +[/3 ?, 
a value very different from that in the former case. 

427. The Composition of Vectors. 

Let an object x be conveyed to y by the operation of a vector, and let the 
object y be then conveyed to z by the operation of a second vector, which we 
shall first suppose to be homonymous (i.e. both right or both left) with 
the preceding. Then we have, from the first, supposed right 

yi = + #! + /&r 2 + 73.3 + 8x 4 , 



2/4 = - Bx l 



474 THE THEORY OF SCREWS. [427, 

and, from the second vector, 

*i = + ay, + ffy* + yy a + 8 y 4 , 



Substituting for y n y 2 , .Vs. y*, we obtain the following values for z lt z%, z s , z 4 . 
The right vector, a, ft, 7, 8, followed by the right vector, a , /? , 7 , 8 



z a = + aa - pp - 77 - 65 ^ + + a/3 + /3a + 75 - 7 x 2 + ay - fid + ya +dp X 3 + + ad + Py - yp + da x 4 , 



z%= -ap -pa -y8 + dy x l + + aa - pp - yy - 85 a; 2 + a5 + 7 - 7(3 + Sa x 3 + - ay + pS -ya - 



= -a7 + /35 --ya -8/3 a; 1 + - aS - Py + 7/3 - da x. 2 + aa - pp - yy - 88 x 3 + + a/3 + /3a + 78 - dy x 4 , 



5a x 1 + +ay - pd + ya +dp x 2 + - a/3 - /3a - 76 + 67 x 3 + + aa - /3/3 - yy - 55 x 4 . 

The right vector, a, ft , 7 , 8 , followed by the right vector, a, ft, 7, 8 



!= + aa - /3/3 - yy - 55 ^ + + a/3 + /3a - 78 + 87 rc 2 + + a> + /35 + 701 - S/S x 3 + + ad - fiy + 7/3 + 8a x 4 , 



z. 2 = - a/3 - /3a + 78 - 87 x l + + aa - /3/3 - 77 - 55 x 2 + + aS - fiy + y{? + 8a x 3 + - ay ~ /35 - ya + 5/3 x t , 



~ 7* + 8/3 ^i + - a5 + fty - 7/8 - 5a J 2 + + aa - /3/3 - 77 - 55 x 3 + + a/3 + /3a - 78 + 87 x 4 , 



yp -8a x 1 + + ay + /36 + ya - 5/3 x. 2 + - a/i - /3a + 75 - 57 x 3 + + aa - /3/3 - 77 - 55 x . 

We thus learn the important truth, that when two or more homonymous 
vectors are compounded, the order of their application must be carefully 
specified. For example, if the object x be first transposed by the vector a 
and then by a , it attains a position different from that it would have gained 
if first transposed by a! and then by a. 

We see, however, that in either case two homonymous vectors compound 
into a vector homonymous with the two components. 

We now study two heteronymous vectors, i.e. one right and one left. 
The right vector, a, ft, 7, 8, followed by the left vector, a , ft , 7 , 8 



z l = +aa -pp -yy -88 x 1 + + a/3 + pa + yd - dy x 2 + + ay - p8 + ya + 5/3 x 3 + +aS + py - 7/8 + 5a z 4 , 



+ aa - /3/3 + 77 + 55 x 2 + -a8 -py - yp + da x 3 + +ay - pd -ya - dp 



= -a7 -/35 -7a + S/3 a; 1 + + aS - pS - yp - da x 2 + + aa + pp - yy + 55 x 3 + - a/3 + /3a - 78 - 8y x 4 , 



Z 4 - - a5 + 7 - 7/3 - Sa x l + - ay - pd + ya - dp x 2 + + a/3 - pa - 78 - 87 x 3 + + aa + pp 1 + yy - 55 x 4 . 

The left vector, a , ft , 7 , 8 , followed by the right vector, a, ft, 7, 8 



! = + aa - pp - 77 - 85 x 1 + -f a/3 7 + /3a + 78 - 87 x 2 + + 07 - /3S + ya + 8p x s + + aS + Py - yp + da x t , 



2 = - ap - pa + yd - dy X 1 + + aa - pp + yy 4 58 X 2 + - ad - Py - yp + da X 3 + + ay - pd - ya - dp X 4 , 



Z 3= -ay -pd -ya + dp x l + + ad - py - yp - da X 2 + + aa + /3/3 - 77 + 55 X 3 + - a/3 + )3a - 76 - 



z 4 = - ad + Py - yp - da X-L+ - ay - pd + ya - dp x 2 + +ap - pa -yd - dy x 3 + + aa + pp + yy - 88 z 4 . 



428] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 475 



We thus learn the remarkable fact, that if a right (left) vector be followed 
by a left (right) vector, the effect produced is the same as if the order of 
the two vectors had been interchanged. 

This is not true for two right vectors or two left vectors. 

The theorems at which we have arrived may be thus generally 
enunciated : 

In the composition of vectors the order of two heteronymous vectors does 
not affect the result, but that of two homonymous vectors does affect the result. 

In the composition of two homonymous vectors the result is also an homony 
mous vector. In the composition of two heteronymous vectors the result is not 
a vector at all. 

The theorems just established constitute the first of the fundamental 
principles relating to the Theory of Screws in non-Euclidian Space referred 
to in 396. Their importance is such that it may be desirable to give a 
geometrical investigation. 

428. Geometrical proof that two Homonymous Vectors com 
pound into one Homonymous Vector. 

Left vectors cannot disturb any 
right generators of the infinite quad- 
ric. Take two such generators, AB 
and A H (Fig. 48). Let AA , 
BB , CC be three left generators 
which the first vector conveys to 
A^Ai, -BjjB/, eft-!, and the second 
vector further conveys to A 2 A 2 , 
B,B 2 , C 2 C 2 . Let X and Y be the 
double points of the two homo- 
graphic systems defined by A, B, G 
and A 2 , B 2 , (7 2 . Then we have 



and 



As anharmonic ratios cannot be 
altered by any rigid displacement, it 
follows that X and Y must each 
occupy the same position after the 
second vector which they had before 
the first, similarly, X and Y will 
remain unchanged, and as the two 
rays, AB and A B are divided homo- lg 8 




476 THE THEORY OF SCREWS. [428- 

graphically, it follows that XX and YY are both generators. We there 
fore find that after the two vectors all the right generators remain as before, 
and so do also two left generators, i.e. the result of the two vectors could 
have been attained by a single vector homonyrnous therewith. 

429. Geometrical proof of the Law of Permutability of Hetero- 
nymous Vectors. 

Let AB and A B be a pair of right generators (fig. 49), and CD and 
C U a pair of left generators. Let the right vector convey P to Q, and 




Fig. 49. 

then let the left vector carry Q to the final position P . We shall now show 
that P would have been equally reached if P had gone first to R, so that 
intervene PR = QP , and that then R was conveyed by the vector, RB B, 
through a distance equal to PQ. 

Draw through P the transversal PRCC . Take R, so that 



but, because this relation holds, 

PQ, RP , CD, G D 
must all lie on the same hyperboloid. 

Therefore RP must intersect AL and AM, and therefore, also, 

(PQAA ) = (RP BB }. 
Hence, finally, we have for the intervenes 

PQ = RP and PR = QP . 

430. Determination of the Two Heteronymous Vectors equi 
valent to any given Motor. 

If a right vector, a, ft, <y, B, be followed by a left vector, a, ft , 7 , 8 , then 



430] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 477 

the result obtained is a displacement of the most general type called a 
motor. We now prove Clifford s great theorem that a right vector and a 
left vector can be determined so as to form any motor, i.e. to accomplish 
any required homographic transformation that conserves intervene. 

For, if we identify the several coefficients at the foot of p. 474 with those 
of 420, we obtain equations of the type 

(11)= ace - ftp - 77 - 88 , 
(21) = -a/9 -/9a + 78 -87 , 
(31) = - ay - 8 - ya + 8/3 , 
(41) = - 08 + /3y - 7/3 - 8a . 

These can be simply reduced to a linear form ; for multiply the first by a , 
and the second, third, and fourth by ft, 7 , 8 , respectively, and add, 
we obtain 



for a 2 + /3 /2 + 7 2 -f 8 2 =1. 

In a similar manner we obtain a number of analogous equations, which 
are here all brought together for convenience 

(11) a! - (21) ft - (31) 7 - (41) 8 = a, 

- (21) a - (11) ft + (41) 7 - (31) 8 = & 

- (31) a - (41) ft - (11) 7 + (21) 8 = 7, 

- (41) a + (31) ft - (21) 7 - (11) 8 = 8. 
+ (22) a! + (12) ft - (42) 7 + (32) 8 = a, 
+ (12) a - (22) ft - (32) 7 - (42) 8 - ft, 



- (32) a! - (42) ft - (12) 7 + (22) 8 = 8. 
+ (33) a + (43) ft + (13) 7 - (23) 8 = a, 
+ (43) a + (33) ft - (23)7 - (13)8 = & 

- (13) a - (23) ft - (33) 7 - (43) 8 = 7, 
+ (23) a! + (13) ft - (43) 7 + (33) 8 = 8. 
+ (44) a - (34) /3 + (24) 7 + (14)8 = a, 
+ (34) a! + (44) ft + ( 14) 7 - (24) 8 = & 

- (24) a - (14) ft + (44) 7 - (34) 8 = 7, 
+ (14) a! - (24) ft - (34) 7 - (44) 8 = 8. 

These will enable a, ft, 7, 8 and a , ft, 7 , 8 to be uniquely determined. 



478 THE THEORY OF SCREWS. [431, 

431. The Pitch of a Motor. 

Any small displacement of a rigid system in the content can be produced 
by a rotation (see 417) a about one line followed by a rotation ft about its 
conjugate polar with respect to the infinite quadric, the amplitudes of both 
rotations being small quantities. The two movements taken together 
constitute the motor. It will be necessary to set forth the conception in 
the theory of the motor, which is the homologue of the conception of pitch 
in the Theory of Screws in ordinary space. The pitch can most conveniently 
be expressed by the function 

2ct/3 
P ~a? + 0*- 

If either a. or ft vanish, then the pitch becomes zero. The motor then 
degenerates to a pure rotation about one or other of the two conjugate 
polars. This, of course, agrees with the ordinary conception of the pitch, 
which is zero whenever the general screw motion of the rigid body degrades 
to a pure rotation. 

In ordinary space we have 

pa. = dft, 

where ft is zero and where d is infinite. In this case 



"a~ d 
i.e. the pitch is proportional to the function now under consideration. 

No generality will be sacrificed by the use of a single symbol to express 
the pitch. We may make a = cos# and /3 = sin#; the pitch then assumes 
the very simple form sin 26. We thus see that the pitch can never exceed 
unity. 

If the motor be a vector, then we have ft = a, or 6 = 45, and the 
pitch is simply + 1. 

It should be noticed that a rotation a about the line A, and a rotation 
ft about its conjugate polar B, constitute a motor of the same pitch as a 
rotation ft about A and a about B. 

432. Property of Right and Left Vectors. 

To take the next step it will be necessary to discuss some of the relations 
between right and left vectors. A right vector will displace any point P 
in a certain direction PA ; a left vector will displace the same point in the 
direction PB. It will, of course, usually happen that the directions PA 
and PB are not identical. It is, however, necessary for us to observe 






432] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 479 

that when P is situated on either of two rays then the directions of dis 
placement are identical. To determine these two rays we draw the two 
pairs of generators corresponding to the two vectors. As these generators 
belong to opposite systems, they will form four edges of a tetrahedron. 
The two remaining edges are a pair of conjugate polars, and they form the 
two rays of which we are in search. The proof is obvious : a point P on 
one of these rays must be displaced along the same ray by either of the 
vectors, for this ray intersects both of the generators which define that 
vector. 

Let a right vector consist of rotations + a, + a about two conjugate polars, 
and let a left vector consist of rotations + a, a, also about two conjugate 
polars. Without loss of generality we may take the two conjugate polars 
in both cases to be the pair just determined. 

Let 00 and PP be two conjugate polars (fig. 50). The right vector is 
appropriate to the generators OP and O P. The left vector to the generators 




Fig. 50. 

OP and O P. If we take the intersections with the quadric in the order 00 
for A, then we must take them on B in the order PP if we are considering 
a right vector, and in the order P P if we are considering a left vector. This 
is obvious, for in the first case we take the intersections of the conjugate 
polar with the generators OP and O P , In the second case we take the 
intersection of the conjugate polar with OP and O P. 

If, therefore, the vector be right, we have for the displacements of X 
and Y, 

H log (XX 00 ) = # log ( YY PP). 



480 THE THEORY OF SCREWS. [432- 

If, however, the vector be left, then Y must be displaced to a distance F , 

defined by 

H log (XX 00 ) = H log (FF P P) : 

we therefore have 

H log ( YY PP ) = Hlog( FF P P) ; 
but from an obvious property of the logarithms, 

H log ( FF P P) = - H log ( FF PP ) ; 
whence, finally, 

H log ( YY PP ) = - fl" log ( FF PP ). 

We hence have the important result, that the intervene through which a 
point on one of the common conjugate polars is displaced by one of two 
heteronymous vectors of equal amplitude, merely differs in sign from the 
displacement which the same point would receive from the other vector. 

433. The Conception of Force in non-Euclidian Space. 

In ordinary space we are quite familiar with the perfect identity which 
subsists between the composition of small rotations and the composition of 
forces. We shall now learn that what we so well know in ordinary space is 
but the survival, in an attenuated form, of a much more complete theory 
in non-Euclidian space. We have in non-Euclidian space force-motors and 
force-vectors, just as we have displacement-motors and displacement-vectors. 
We shall base the Dynamical theory on an elementary principle in the 
theory of Energy. Suppose that a force of intensity / act on a particle 
which is displaced in a direction directly opposed to the force through a 
distance 8, then the quantity of work done is denoted by fS. 

434. Neutrality of Heteronymous Vectors. 

We are now able to demonstrate a very important theorem which lies at 
the foundation of all the applications of Dynamics in non-Euclidian space. 
The virtual moment of a force-vector and a displacement-vector will always 
vanish when the vectors are homonymous and at right angles. The analogies 
of ordinary geometry would have suggested this result, and it is easily shown 
to be true. If, however, the two vectors be not homonymous, the result is 
extremely remarkable. The two vectors must then have their virtual moment 
zero under all circumstances. 

The proof of this singular proposition is very simple. Let the two 
vectors be what they may, we can always find one pair of conjugate polars 
which belong to them both. Let the two forces be X, A, on the two conjugate 
polars, and let the displacements be p, p, then the work done is 

X/4 XyU, = 0. 



434] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 481 

It seems at first sight incredible that a theorem demonstrated with such 
simplicity should be of so much significance. It is not too much to say 
that the theory of Rigid Dynamics in non-Euclidian space depends, to a 
large extent, upon this result. This is the second of the two fundamental 
theorems referred to in 396. 

These two principles open up a geometrical Theory of Screws in non- 
Euclidian space. This is a subject too extensive to be here entered into any 
further. It is hoped that the present chapter will at least have conducted 
the reader to a point from which he can obtain a prospect of a great field of 
work. The few incursions that have as yet been made into this field (see the 
bibliographical notes) have shown the exceeding richness and interest of a 
region that still awaits a more complete exploration. 



31 



APPENDIX I. 

NOTE I. 

Another solution of the problem of 28. 

LET the intensities of the wrenches on a, ft, ... rj be as usual denoted by 
a", ft", ... 17" respectively. 

As the wrenches are to equilibrate we must have ( 12) 



where A is any screw whatever. 

If six different but independent screws be chosen in succession for A we have 
six independent linear equations, and thus a" ft" and the other ratios are known. 

But the process will be much simplified by judicious choice of A. If, for 
instance, we take as A the screw if/ which is reciprocal to the five screws y, 8, e, , rj 
then we have 

tff y ^ = 0, t3- 5l ^ = 0, HT^ = 0, -ET^ = 0, Sr r)v j, = 0, 

and we obtain 

a"w al j, + ft" -or w = 0. 

Let p be a screw on the cylindroid defined by a and ft. Then wrenches on 
a, ft, p will equilibrate ( 14) provided their intensities are proportional re 
spectively to 

sin (ftp), sin (pa), sin (a/3). 
It follows that for any screw p. we must have 

sin (ftp) CT aM + sin (pa) OT/Jjll + sin (aft) cr p/ot = 0. 

This is indeed a general relation connecting the virtual coefficients of three 
screws on a cylindroid with any other screw. 

Let us now suppose that p. was the screw ^ just considered, and let us further 
take p to be that one screw on the cylindroid (a, ft) which is reciprocal to if/. 

Then 

Tp* = 0, 

and we have 

sin (ftp) iy a<il + sin (pa) CT^ = 0. 

31-2 



484 THE THEORY OF SCREWS. 

But since the seven screws are independent both C7 a ^ and trr^ must be, in 
general, different from zero, whence by the former equation we have 

a" ft" 

sin (ftp) sin (pa) 

Thus we obtain the following theorem ( 28). 

If seven wrenches on seven given screws equilibrate and if the intensity a" of 
one of the seven wrenches be given then the intensity of the wrench on any one 
ft of the remaining six screws can be determined as follows. 

Find the screw ty reciprocal to the five screws remaining when a and (3 are 
excluded from the seven. 

On the cylindroid (aft) find the screw p which is reciprocal to i/^. 

Resolve the given wrench a" on a into component wrenches on ft and on p. 

Then the intensity of the component wrench thus found on ft is the required 
intensity ft" with its sign changed. 



NOTE II. 

Case of equal roots in the Equation determining Principal Screws of 

Inertia, 86. 

We have already made use of the important theorem that if U and V are both 
homogeneous quadratic functions of n variables, then the discriminant of U + A V 
when equated to zero must have n real roots for X provided that either U or V 
admits of being expressed as the sum of n squares ( 85). 

The further important discovery has been made that whenever this deter- 
minantal equation has a repeated root, then every minor of the determinant 
vanishes (Routh, Rigid Dynamics, Part II. p. 51, 1892). 

This theorem is of much interest in connection with the Principal Screws of 
Inertia. The result given at the end of 86 is a particular case. It may be 
further presented as follows. 

Taking the case of an n system each root of A. will give n equations 

l dT ! dT 



Of these n - 1 are in general independent and these suffice to indicate the values 
of *,...*,,. 

But m the case of a root once repeated the theorem above stated shows that we 
have not more than n - 2 independent equations in the series. The principal 
Screw of Inertia corresponding to this root is therefore indeterminate. 

But it has a locus found f rom the consideration that besides these u - 2 linear 



APPENDIX I. 485 

equations it must also satisfy the 6 n linear equations which imply that it belongs 
to the ?z-system. 

In other words the co-ordinates satisfy (n - 2) + (6 n), i.e. four linear equations. 
But this is equivalent to saying that the screw lies on a cylindroid. We have thus 
the following result. 

In the case when the equation for X has two equal roots, there must be n 2 
separate and distinct principal screws of inertia and also a cylindroid of which 
every screw is a principal Screw of Inertia. 

For every value of n from 1 to 6 it is of course known that the celebrated 
harmonic determinantal equation of 86 has n real roots. 

But when the question arises as to the possibility of this equation as applied 
to our present problem having repeated roots, the several cases must be dis 
criminated. It is to be understood that the body itself is to be of a general type 
without having e.g. two of the principal radii of gyration equal. The investigation 
relates to the possibility of a system of constraints which, while the body is still of 
the most general type, shall permit indeterminateness in the number of principal 
Screws of Inertia. 

Of course if n - 1 the equation is linear and has but a single root. 

n 2. The equation under certain conditions may have two equal roots. 

n = 3. The equation under certain conditions may have two or three equal 
roots. 

n = 4. The equation under certain conditions may have two equal roots or 
three equal roots or four equal roots or two pairs of equal roots. 

n 5. The equation can never have a repeated root. 
n = 6. The equation can never have a repeated root. 

Here comes in the restriction that the body is of a general type, for of course 
the last statement could not be true if two of the radii of gyration are equal or if 
one of them was zero. 

The curious contrast between the two last cases and those for the smaller 

^7, Ifn 4- 1 ^ 

values of n may be thus accounted for. The expression for T will contain ^s - 

a 

terms and the ratios only being considered T will contain 

n(n+l) _ _ (n + 2)(n-l) 

~2~ ~1T~ 

distinct parameters. As a rigid body is specified both as to position and character 
by 9 co-ordinates, it follows that the coefficients of T are not unrestricted if 

- is greater than 9. But this quantity is greater than 9 for the cases 

2i 

of n 5 and n = 6. 

We may put the matter in another way which will perhaps make it clearer. 
I shall take the two cases of n = 4 and n = 5. 



486 THE THEORY OF SCREWS. 

In the case of n -- 4 the function T will consist of 10 terms such as 

If any arbitrary values be assigned to A u , A }a &c., it will still be possible to 
determine a rigid body such that this function shall represent u e 2 (to a constant 
factor), because we have 9 co-ordinates disposable in the rigid body. Hence for 
n - 4 and a fortiori for any value of n less than four the function representing T 
will be a function in which the coefficients are perfectly unrestricted. Hence 
n = or < 4 the determinantal equation is in our theory of the most general type. 
The general theory while affirming that all the roots are real does not prohibit 
conditions arising under which roots are repeated. Hence Routh s important 
theorem becomes of significance in cases n=2, n 3, n = 4: for in these equations 
the roots may be repeated. 

But in the case of n - 5 the function T consists of 1 5 terms. If arbitrary 
values could be assigned to the coefficients then of course the general theory would 
apply and cases of repeated roots might arise. But in our investigation the 15 
coefficients are functions of the co-ordinates which express the most general place 
of a rigid body, and these co-ordinates are not more than nine. If these nine co 
ordinates were eliminated we should have five conditions which must be satisfied 
by the coefficients of a general function before it could represent the T of our 
theory even to within a factor. The necessity that the coefficient of T shall satisfy 
these equations imports certain restrictions into the general theory of the deter 
minantal equation based on T. One of these restrictions is that T shall have no 
repeated roots. The same conclusion applies a fortiori to the case of n = 6. 

The subject may also be considered as follows. 

Let us first take the general theorem that when reference is made to n 
principal screws of inertia of an n-system the co-ordinates of the impulsive wrench 
corresponding to the instantaneous screw 

are ( 97) 

^a,.-- X 

Pi Pn 
For a principal screw of inertia the ratios must be severally equal or 

Pi <*i Fa "2 Pn a-n 

These equations can generally be only satisfied if n - 1 of the quantities 

be zero, i.e. there are in general no more than the n principal screws of inertia. 
If however 

Pi P-2 

then though we must have 

a, = 0...a w = 0, 
aj and a 2 remain arbitrary. 



APPENDIX I. 487 

But n - 2 linear equations in an -re-system determine a cylindroid and hence we 
see that all the screws on this cylindroid will be principal Screws of Inertia. 

In like manner if there be k repeated roots, i.e. if 



Pi Pa >* 

then a 1} ... a k are arbitrary but a fc+1 , ... a n must be each zero. We have thus n k 
linear equations in the co-ordinations. They must also satisfy 6 - n equations 
because they belong to the %-system and therefore they satisfy in all 

Q-n + n k=6k equations, 
whence we deduce that 

If there be k repeated roots in the determinantal equation of 86 then to those 
roots corresponds a k-system of screws each one of tvhich is a principal screw of 
inertia and there are besides n -k additional principal Screws of Inertia. 

So far as the cases of n = 2 and n = 3 are concerned the plane representations 
of Chaps. XII. and XV. render a complete account of the matter. 

Let (Fig. 10) be the pole of the axis of pitch, 58, then may lie either 
inside or outside the circle whose points represent the screws on the cylindroid. 

Let (Fig. 22) be the pole of the axis of inertia, 140, then must lie inside 
the circle, for otherwise the polar of would meet the circle, i.e. there would 
be one or two real screws about which the body could twist with a finite velocity 
but with zero kinetic energy. 

We have seen that the two Principal Screws of Inertia are the points in which 
the chord 00 cuts the circle. If could be on the circle or outside the 
circle then we might have the two principal Screws of Inertia coalescing, or we 
might have them both imaginary. As however must be within the circle it is 
generally necessary that the two principal Screws of Inertia shall be both real and 
distinct. 

But the points and might have coincided. In this case every chord through 
would have principal Screws of Inertia at its extremities. Thus every point on 
the circle is in this case a principal Screw of Inertia. 

We thus see that with Freedom of the second order there are only two possible 
cases. Either every screw on the cylindroid is a principal Screw of Inertia or 
there are neither more nor fewer than two such screws, and both real. 

If a and /3 be any two screws on the cylindroid then the conditions that all 
the screws are Principal Screws of Inertia are 



With any rigid body in any position we can arrange any number of cylindroids 



488 THE THEORY OF SCREWS. 

which possess the required property. Choose any screw a and then take any screw 
ft whose co-ordinates satisfy these two conditions. 

We shall also use the plane representation of the 3-system. 
Let U = be the pitch conic. 

V = be the imaginary ellipse obtained by equating to zero the expression for 
the Kinetic Energy. 

Then the vertices of a common conjugate triangle are of course the principal 
Screws of Inertia and generally there is only one such triangle. 

It may however happen that U and V have more than a single common 
conjugate triangle, for let the cartesian co-ordinates of the four intersections of U 
and V be represented by 

X lt 111. ) X 2) 2/2J X 3) 2/3J a 4> 2/4 

As all the points on V are imaginary at least one co-ordinate of each intersection 
is imaginary. Suppose y l to be imaginary then it must be conjugate to y. 2 . If 
therefore the conic U touches V y l and y. A must be respectively equal to y% and y 4 . 
Hence we have only two values of y, and these are conjugate. Substituting these 
in U and V we see that there can only be two values of x, and consequently the 
intersections reduce to two pairs of coincident points. 

Hence we see that V cannot touch U unless the two conies have double contact. 

In this case the chord of contact possesses the property that each point on it is 
a principal Screw of Inertia while the pole of the chord with respect to either 
conic is also a principal Screw of Inertia. 

If U and V coincided then every screw of the 3-system would be a principal 
Screw of Inertia. 

The general theory on the subject is as follows. 

Let (7= be the quadratic relation among the co-ordinates of an ?t-system 
which expresses that its pitch is zero. 

Let V = be the quadratic relation among the co-ordinates of a screw if a body 
twisting about that screw has zero kinetic energy. 

The discriminant of S = U + A V equated to zero gives n real roots for A. These 
roots substituted in the differential coefficients of S equated to zero give the 
corresponding principal Screws of Inertia. If however there be two equal roots 
for X then for these roots every first minor of the discriminant vanishes. In this 
case S can be expressed as a function of n - 2 linear quantities. Perhaps the most 
explicit manner of doing this is as follows. 

Let S = a u 6f + a^O* + 2a,AO, + . . . + a nn O n \ 

and let ,-!^ -I !*? 



APPENDIX I. 



489 



If all the first minors of the discriminant of S vanish we must have the 
following identity 



nn n > 



by which we have 

S = A 33 s 3 2 + A^s? + 2A u s 3 s 4 ... 4- A nn s n 2 . 

Hence U + XV = A^s.* + ^ 44 s 4 2 + 2^348^4 . . . + A nn s n 2 . 

In the case of n = 3 we have 



which proves that V and U have double contact as we already proved in a different 
manner. 

In the general case all the differential coefficients of S will vanish if s 3 =Q...s n ~0, 
but these latter define a cylindroid and therefore whenever the discriminant of s 
has two equal roots, every screw on a certain cylindroid is a principal Screw of 
Inertia. 

If the discriminant had three equal roots then S could be expressed in terms of 
s 4 , ...s n and in this case every screw on a certain 3-system would be a principal 
Screw of Inertia. 

If n 1 of the roots of the discriminant were equal, then every (n 2)nd 
minor would vanish, S would become the perfect square s n 2 to a factor. 

And we have 



In this case every screw of the n - 1 system defined by s n = will be a principal 
Screw of Inertia. 



NOTE III. 

Twist velocity acquired by an impulsive wrench, 90. 

The problem solved in 90 may be thus stated. 

A body of mass M only free to twist about a. is acted upon by a wrench of 
intensity tf" on a screw rj. Find the twist velocity acquired. 

From Lagrange s equations we have, 86 

d fdT\ dT 
- = 



490 THE THEORY OF SCREWS. 

But as the wrench is very great the initial acceleration is great and conse 
quently the second term on the left-hand side is negligible compared with the first. 

dT C 

Whence -^ = 2^ a \ifdt = 2isr 1?a i 7 " ; , 

CX/CL J 

but T=Mu a *a*(89), 

whence 2Mit a 2 a=2 ar ria r) ", 

or " = 77 "2 V- 

The kinetic energy 

T=Mu^ = ^^ "\ (91). 



NOTE IV. 

Professor C. J. Jolys theory of the triple contact of conic and cubic. 

Professor C. J. Joly has pointed out to me that the conies of 162 and 165 
which have triple contact with the nodal cubic are but particular instances of the 
more general theory which he investigates as follows. 

Let t be the parameter necessary to define a particular generator on a given 
cylindroid ; we first show that the condition that a line, i.e. a screw of zero pitch 
should intersect this generator may be expressed in the form 

at 3 + W + ct + d = 0, 

where a, b, c, d are linear functions of the co-ordinates of and, of course, functions 
also of the constants defining the cylindroid. 

For if a and ft be the