512 THE THEORY OF SCREWS. 



MOBIUS (A. F.) Ueber die Zusammensetzung unendlick 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 Bittershaus, p. 524). 



RODRIGUES (O.) Des lois geometriques qui regissent les deplacements d un systeme 

 solide dans Vespace, et de la variation des coordonnees prorenant de ces 

 deplacements consideres independamment des causes qui peuvent les produire. 

 Liouville s Journal Math. ; Vol. v., pp. 380440 (5th Dec., 1840). 



This paper consists mainly of elaborate formulae relating to displacements of 

 finite magnitude. It has been already cited for an important remark ( 9). 



CHASLES (M.) Proprieties geometriques relatives au mouvement infiniment petit 

 dans un corps solide libre dans I espace. Paris, Comptes Rendus ; Vol. xvi., 

 pp. 1420-1432 (1843). 



A pair of &quot;droites conjuguees&quot; are two lines by rotations about which a given 

 displacement can be communicated to a rigid body. Two pairs of &quot;droites con 

 juguees&quot; 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. (18451847). 

 Appendix No. 5, pp. li. Ix. (Communicated Dec. 8, 1845.) 



On p. Ivii. he states &quot;the laws of equilibrium of several forces applied to various 

 points of a solid body, are thus included in the two equations, 



2/3 = 0; 2 (a0 - 0a) = ; 



the vector of the point of application being a, and the vector representing the 

 force applied at that point being /3.&quot; On the same page he writes, 



&quot;Instead of the two equations of equilibrium, we may employ the single 

 formula 



2 . aft = - 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.&quot; 



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 



2 Fa/3 



- = w 



