A Treatise on the Theory of Screws

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. Pi-operty 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 Formulæ 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 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...................................................306 289. Construction of Chiastic Homography on the Cylindroid .... 307 290. Hoinographic Systems on Two Cylindroids...............................307 291. Case of Normal Cylindroids............................................308 292. General Conditions of Chiastic Homography.............................309 293. Origin of the Formulæ of § 281........................................310 294. Exception to be noted.................................................312 295. Impulsive and Instantaneous Cylindroids...............................312 296. An Exceptional Case...................................................314 297. Another Extreme Case..................................................316 298. Three Pairs of Correspondents.........................................317