148 On the Twenty-seven Lines on a Cubic Surface. [June 15, 



looked that a displacement is not defined by the direction of axis, 

 and amplitude, of the resultant rotation, together with the magnitude 

 of the component of the corresponding translation along that direc- 

 tion (for in this form the proof is given, the axis being drawn through 

 one end of the common perpendicular to the particular couple in 

 respect of which the theorem is demonstrated), since these elements 

 are common to an infinity of displacements. 



This being premised, the laws connecting pairs of axes by successive 

 rotations round which a given displacement of a rigid body in space 

 may be effected are as follows : 



If the first axis (* g) is taken arbitrarily, say parallel to a given 

 vector, ', and passing through the term of a second given vector, ", its 

 conjugate is parallel to a vector (), the side common to three quadric 

 cones, the constants of which are functions of ', ', and the vectors 

 defining the displacement. 



Each of these cones, whatever the direction of "*, passes through 

 one of three fixed vectors. 



The directions of the axes being fixed in accordance with the above 

 conditions, the locus of either axis is a plane, the places of the axes 

 in which are so related that the connector of the feet of perpen- 

 diculars on them from any fixed point generates a ruled quadric 

 surface. 



[The last three paragraphs have been altered (July 15) after a 

 correspondence, since the reading of the note on 15th June, with 

 which Professor W. Burnside, F.R.S. (who, however, is not respon- 

 sible for any statement herein), favoured me; as the result of 

 which he sent me a geometrical proof that one axis might in all cases 

 be taken arbitrarily both in position and direction. On revising my 

 analysis, I found that what I had taken as an equation of condition 

 was reducible to an identity.] 



XI. "On a Graphical Representation of the Twenty-seven 

 Lines on a Cubic Surface." By H. M. TAYLOR, M.A., 

 Fellow of Trinity College, Cambridge. Communicated by 

 A. R. FOURTH, Sc.D., F.R.S. Received June 13, 1893. 



(Abstract.) 



The converse of Pascal's well-known theorem may be stated thus : 

 if two triangles be in perspective, their non-corresponding sides 

 intersect in six points tying on a conic. An extension of this theorem 

 to three dimensions may be stated thus : if two tetrahedrons be in 

 perspective, their non-corresponding faces intersect in twelve straight 



