520 THE THEORY OF SCREWS. 



BALL (R. S.) On the small oscillations of a Riyid Body about a fixed point under 

 tlie action of any forces, and, more particularly, when gravity is the only 

 force acting. Transactions of the Royal Irish Academy; Vol. xxiv., pp. 

 593-628 (January 24, 1870). 



Certain dynamical problems which are here solved for the rotation of a body 

 round a point were solved in subsequent papers for a body restricted in any 

 manner whatever. Some of the chief results obtained are given in 197. 



This paper has its geometrical basis in the following theorem, due apparently to 

 D Alembert. Recherches sur la Precession des Equinoxes, Paris, 1749, p. 83. 



Any small displacement of a rigid body rotating around a fixed point can 

 be produced by the rotation around an axis passing through the point. 



In 1776 Euler proved that the same law was true for displacements of finite 

 magnitude. Formulae generales pro translations, quacunque corporum Rigidorum. 

 Novi Commentarii Academiae Petropolitanae ; Vol. xx., pp. 189-207. 



KLEIN (F.) Notiz betreffend den Zusammenhang der Liniengeometrie mit der 

 Mechanik starrer Korper. Math. Ann.; Vol. iv., pp. 403-415 (June, 1871). 



Among many interesting matters this paper contains the germ of the physical 

 conception of reciprocal screws. We thus read on p. 413: &quot;Es lasst sich nun in 

 der That ein physikalischer Zusammenhang zwischen Kraftesystemen und unend- 

 lich kleinen Bewegungen angeben, welcher es erklart, wie so die beiden Dinge 

 mathematisch co-ordinirt auftreten. Diese Beziehung ist nicht von der Art, dass 

 sie jedem Kraftesystem eine einzelne unendlich kleine Bewegung zuordnet, sondern 

 sie ist von anderer Art, sie ist eine dualistiche. 



&quot;Es sei ein Kraftesystem mit den Coordinaten E, H, Z, A, M, N, und eine 

 unendlich kleine Bewegung mit den Coordinaten H , // , Z , A , M , N gegeben, 

 wobei man die Co-ordinaten in der im 2 besprochenen Weise absolut bestimmt 

 haben mag. Dann reprdsentirt, wie hier nicht weiter nachgewiesen werden soil, 

 der Ausdruck 



A H + M N + S Z + H A + H M + Z N 



das Quantum von Arbeit, welches das gegebene Kraftesystem bei Eintritt der 

 gegebenen unendlich kleinen Bewegung leistet. Ist insbesondere 



A H + M U + N Z + H A + H M H- Z N= 0, 



so leistet das gegebene Kraftesystem bei Eintritt der gegebenen unendlich kleinen 

 Bewegung keine Arbeit. Diese Gleichung nun reprasentirt uns, indem wir einmal 

 H, H, Z, A, M, N, das andere H , // , Z , A , M , N als veranderlich betrachten, den 

 Zusammenhang zwischen Kraftesystemen und unendlich kleinen Bewegungen.&quot; 



BALL (R. S.) The Theory of Screws a geometrical study of the kinematics, 

 equilibrium, and small oscillations of a Riyid Body. First memoir. Trans 

 actions of the Royal Irish Academy, Vol. xxv., pp. 137-217 (November 13, 

 1871). 



This is the original paper on the Theory of Screws. At the time this paper 

 was printed (1871) I had no suspicion that the Cylindroid had been ever studied 

 by anyone besides myself. I subsequently learned that the same surface had 

 been investigated by Pliicker two or three years previously (1868-9) in connection 

 with the linear complex (see pp. 20, 517). It also appeared that about the same 

 time (1869) this surface presented itself in the Researches of Battaglini. Indeed, 

 to this mathematician belongs, I believe, the distinction of having been the first to 

 perceive that this particular conoid had a special dynamical significance (see 

 pp. 20, 518). Pliicker and Battaglini were certainly independent discoverers of the 



