1894.] Prof. Ball, Note on Geometrical Mechanics. 241 



It appears to be of interest to find the geometrical connection 

 between each screw about which the body can twist and the 

 corresponding restraining screw. We proceed thus. 



The screws of the three systems can be represented by the 

 several points in a plane. All the points on one conic (I.) corre- 

 spond to screws of zero pitch; all the points on another conic (II.) 

 correspond to screws about which the body would twist with zero 

 kinetic energy (of course this is imaginary). Let P be any screw, 

 draw the polar of P with regard to II., then Q, the pole of this 

 ray with regard to I., indicates the screw, an impulsive wrench on 

 which would make the body commence to twist about P. 



There is a certain homography in the plane such that if P' be 

 the correspondent of P, then the pole of the ray PP' with regard 

 to I. points out the restraining screw, corresponding to P, while 

 the pole of PP' with regard to II. indicates the screw, a twist on 

 which is the acceleration of the body twisting around P. 



The three double points of this homography are the points 

 corresponding to the three permanent screws, i.e. those on which 

 the body twists without any immediate tendency to depart. 



(2) On the Construction of a model of 27 Straight Lines upon 

 a Cubic Surface. By W. H. Blythe, M.A., Jesus College, 

 Cambridge. 



Every cubic surface contains twenty-seven straight lines real 

 or imaginary, these lie in forty-five planes, in sets of three, there 

 are consequently one hundred and thirty-five points of intersection 

 of these lines. 



Examining the position of these points two things are evident, 

 first, that ten of them lie in each of the twenty-seven straight 

 lines, and secondly, that since any two of the planes intersect in a 

 straight line, the straight lines forming any two independent 

 triangles intersect in pairs in three points lying in a straight 

 line. 



It is clear therefore that by taking sufficient points to deter- 

 mine the constants in the general equation, we can at the same 

 time by means of these points, find the remainder of the hundred 

 and thirty-five by simple geometrical construction. 



It is shewn (Art. 3) that seven points of intersection can be 

 so taken as to determine all constants in the general equation 

 given by Taylor, and that these also are sufficient to find all 

 remaining points of intersection. 



