Ball — Dynamical Problems in the Theory of a Rigid Body. 29 



III. — 0^ A Plane Representation of Certain Dynamical Problejis 



IN THE THEOEr OF A RiGII) BoDT. By EoBEET S. BaLL. 



[Read, AprU 9, 1883.] 



In the following Paper I propose to exhibit a method of studyiug by 

 merely plane construction the theory of certain problems in a rigid 

 system which has two degrees of freedom. The problems referred to 

 are those in which the rigid body remains always in the vicinity of its 

 original position, so that the dynamical questions are merely those of 

 equilibrium, of impulsive forces and of small oscillations. 



We must refer to the "Theory of Screws" for an outline of the 

 principles on which the present method is based. We there find that 

 when a body has freedom of the second order it is always capable of 

 twisting about every screw of a certain group which lie on the ruled 

 cubic surface called the cylindroid. Each screw has a pitch appro- 

 priate to its situation on the cylindi'oid, so that while the body has two 

 degrees of freedom, and is thus capable of attaining a position which 

 can be defined by two generalizecl co-ordinates, one of these co-ordi- 

 nates may be regarded as indicating the screw about which the body 

 is twisted, while the other gives the amplitude of the twist. We may 

 consider each screw of the system to be denoted by a point in a plane, 

 or we may regard a group of points in a plane which correspond respec- 

 tively with the group of screws on the cylindroid. As the screws on 

 the cylindroid are only a singly infinite series, so the points in the 

 plane must be only a single infinite series ; in other words, they must 

 lie upon a curve. As also when we proceed around the cylindroid we 

 return to the screw from which we started, it would seem natural that 

 the curve of points should be a closed curve. What is this closed curve 

 to be ? It will be easy to show that we can obtain many advantages 

 by taking this curve to be a cii'cle. 



Let a and jB be two screws on the cylindroid. We select first 

 of all a point A quite arbitrarily which shall correspond to a. 

 We may for the moment regard the choice of B, the point corre- 

 sponding to /?, as also arbitrary. Let ^ be a third screw on the cylin- 

 droid, and let P be its corresponding point on the plane. There will 

 be a great convenience in choosing P, so that the triangle ABP shall 

 bear some quickly intelligible relation to the set of screws a, y8, 0. 

 We are here reminded of the fundamental property of the cylindroid 

 that three twists on any three screws will neutralize, provided that 

 each twist has an amplitude proportional to the sine of the angle, 

 between the other two screws. It is therefore natural to choose P, 

 so that the sides of the triangle ABP shall be respectively propor- 

 tional to the amplitudes of the twists about the corresponding screws. 

 Hence it follows that the angle which P subtends at A and B must 

 be equal to the angle between the two screws A and B. Now as this 



