514 THE THEORY OF SCREWS. 



CAYLEY (A.) On a new analytical, representation of curves in space. Quarterly 

 Mathematical Journal; Vol. iii., pp. 225-236 (1860). Vol. v., pp. 81-86 

 (1862). Coll. Math. Papers, Vol. iv. pp. 446-455, 490-494. 



In this paper the conception of the six co-ordinates of a line is introduced for 

 the first time. This is of importance in connection with our present subject 

 because the six coordinates of a screw may be regarded as the generalization of 

 the six coordinates of a straight line. 



If ttj,... a g be the six coordinates of a screw then when we express that its 

 pitch is zero by the condition 



we obtain the coordinates of a straight line. This is perhaps the most symmetrical 

 form of the quadratic condition which must subsist between six quantities consti 

 tuting the coordinates of a line. In Cayley s system it is given by equating the 

 sum of three products to zero. 



SYLVESTER (J. J.) Sur V involution des lignes droites dans I espace, considerees 

 comme des axes de rotation. Paris, Comptes Rendus ; vol. Iii., pp. 741-746 

 (April, 1861). 



Any small displacement of a rigid body can generally be represented by rota 

 tions about six axes (Mobius). But this is not the case if forces can be found 

 which equilibrate when acting along the six axes on a rigid body. The six axes in 

 this case are in involution. The paper discusses the geometrical features of such 

 a system, and shows, when five axes are given, how the locus of the sixth is to be 

 found. Mobius had shown that through any point a plane of lines can be drawn 

 in involution with five given lines. The present paper shows how the plane can 

 be constructed. All the transversals intersecting a pair of conjugate axes are in 

 involution with five given lines. Any two pairs of conjugate axes lie on the same 

 hyperboloid. Two forces can be found on any pair of conjugate axes, which are 

 statically equivalent to two given forces on any other given pair of conjugate axes. 

 In presenting this paper M. Chasles remarks that Mr Sylvester s results lead to the 

 following construction : Conceive that a rigid body receives any small displace 

 ment, then lines drawn through any six points of the body perpendicular to their 

 trajectories are in involution. M. Chasles also takes occasion to mention some 

 other properties of the conjugate axes. 



SYLVESTER (J. J.) Note sur I involution de six lignes dans I espace. Paris, Comptes 

 Rendus; vol. Hi., pp. 815-817 (April, 1861). 



The six lines are 1, 2, 3, 4, 5, 6. Let the line i be represented by the 

 equations 



a t x 4- l&amp;gt; i y + CiZ + diU = 0, 

 a t x + fay + jiZ + o t u - 0, 



and let i, j represent the determinant 



a t b { c t 



* A Ji 

 ttj bj Cj 



a j Pj 7j 8 



