522 THE THEORY OF SCREWS. 



more generally of two Dynames in Pliicker s sense or of two &quot;motors&quot; as Clifford 

 prefers to call them. A &quot; motor&quot; may be said to bear the same relation to a screw 

 which a vector bears to a ray. The calculus of Biquaternions is generalized from 

 that of quaternions and belongs to the non-Euclidian geometry. See Klein s 

 celebrated paper, &quot;Ueber die sogenannte nicht Euclidische Geometric.&quot; Math. 

 Ann., Band IV., pp. 573-G25. This paper of Clifford s has been the commence 

 ment of an extensive theory at which many mathematicians have since worked. 

 Chap xxvi. discusses some of Clifford s theorems and in the course of these 

 bibliographical notes there are several references to this theory. See under the 

 names of Everett, Padeletti, Cox, Heath, Buchheim, Cayley, Burnside, Joly, 

 Kotelnikof, and M Aulay. 



BALL (R. S.) Researches in the Dynamics of a Rigid Body by the aid of the Theory 

 of Screivs. Second Memoir (June 19, 1873). Philosophical Transactions, 

 pp. 15-40 (1874). 



The chief advance in this paper is expressed by the theorem that a rigid body 

 has just so many principal screws of inertia as it has degrees of freedom. This 

 theorem is a generalization for all cases of a rigid system, no matter what be the 

 nature and number of its constraints, of the well-known property of the principal 

 axes of a rigid body rotating around a fixed point. 



It is shown that if the screws on one cylindroid be regarded as impulsive 

 screws, the system of corresponding instantaneous screws lie on another cylindroid. 

 Any four screws on the one cylindroid, and their four correspondents on the others 

 are equi-anharmonic. This theorem leads to many points of connexion between 

 theoretical dynamics and modern geometry. It has been greatly developed sub 

 sequently. 



A postscript to this paper gives a brief historical sketch which shows the rela 

 tion of the theory of screws to the researches of Pliicker and Klein on the Theory 

 of the Linear Complex. 



SKATOW. Zusaminenstellung der Sdtze von den ubriybleibenden Bewegungen eines 

 Kdrpers, der in einigen Punkten seiner oberfldche durch normale Stiitzen 

 unterstiitzt wird. Schlomilch s Zeitschrift fur Mathem. u. Physik, B. xviii., 

 p. 224, 1873. 



HALPHEN Sur le deplacement d line solide invariable. Bulletin de la Soc. Math., 

 Vol. ii., pp. 56-62 (23 July, 1873). 



The study of the displacements of a rigid body is distributed into six cases 

 according to the number of degrees of freedom. This paper like so many others on 

 the present subject has been suggested by the writings of M. Mannheim. It 

 gives for instance a proof of Mannheim s theorem that all the displacements of a 

 solid restrained by four conditions could be produced by two rotations around two 

 determinate lines. These are of course in our language the two screws of zero 

 pitch on the cylindroid expressing the freedom. Halphen considers in some cases 

 conditions more general than those of Mannheim and adds some theorems of quite 

 a new class. Thus still referring to the case of a body restrained by four con 

 ditions, i.e. with two degrees of freedom, he shows how the movements of every 

 point are limited to a surface, and then calling the two screws of zero pitch the 

 &quot;axes&quot; we have as follows. &quot; Les projections, sur un plan donne, des elements 

 superficiels, decrit par les points du corps, sont proportionelles aux produits des 

 segments intercepted, sur des secantes paralleles issues de ces points, par un para- 

 boloide passant par les deux axes, et ayant le plan donne pour plan directeur.&quot; 



