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 



