A Treatise on the Theory of Screws

512 THE THEORY OF SCREWS. Möbius (A. F.)— Ueber die Zusammensetzung unendlich kleiner Drehungen. Crelle’s Journal; Vol. xviii., pp. 189—212 (1838). This memoir contains many very interesting theorems, of which the following are the principal:—Any given small displacement of a rigid body can be effected by two small rotations. Two equal parallel and opposite rotations compound into a translation. Small rotations about intersecting axes are compounded like forces. If a number of forces acting upon a free body make equilibrium, then the final effect of a number of rotations (proportional to the forces) on the same axes will be zero. If a body can undergo small rotations about six independent axes, it can have any small movement whatever. He illustrates this by the case of a series of bodies of which each one is hinged to those on either side of it. If the first of the series be fixed then in general the seventh of the series will be perfectly free for small movements (see Rittershaus, p. 524). Rodrigues (O.)—Des lois géométriques qui régissent les deplacements d’un Systeme solide dans Vespace et de la variation des coordonnées provenant de ces déplacements considérés indépendamment des causes qui peuvent les produire. Liouville’s Journal Math.; Vol. v., pp. 380—440 (5th Dec., 1840). This paper consists mainly of elaborate formulæ relating to displacements of finite magnitude. It has been already cited for an important remark (§ 9). Chasles (M.)—Propriétés géométriques relatives au mouvement infiniment petit dans un corps solide libre dans Vespace. Paris, Comptes Rendus; Vol. xvi., pp. 1420-1432 (1843). A pair of “ droites conjuguées ” are two lines by rotations about which a given displacement can be communicated to a rigid body. Two pairs of “droites con- juguées” are always generators of the same hyperboloid. Hamilton (Sir W. R.)—On some additional applications of the Theory of Algebraic Quaternions. Royal Irish Academy Proceedings; Vol. iii. (1845—1847). Appendix No. -5, pp. li.—lx. (Communicated Dec. 8, 1845.) On p. Ivii. he states “the laws of equilibrium of several forces applied to various points of a solid body, are thus included in the two equations, 2/8=0; s (aß - ßa) = 0 ; the vector of the point of application being a, and the vector representing the force applied at that point being ß.” On the same page he writes, “ Instead of the two equations of equilibrium, we may employ the single formula 2 . aß = - c, c here denoting a scalar (or real) quantity, which is independent of the origin of vectors, and seems to have some title to be called the total tension of the system.” Hamilton (Sir W. R.)—Some applications of Quaternions to questions connected with the Rotation of a Solid Body. Royal Irish Academy Proceedings; Vol. iv. (1847—1850) pp. 38—56. (Communicated Jan. 10, 1848.) In this paper with the same notation as before, he takes the general case of a Rigid Body acted on by forces and considers the Quaternion -^ß = w + ix + jy + kz.