254 Proceedings of the Royal Irish Academy. 



If P be the plane at infinity, Q becomes the " plane of centres." 



Hence the theorems for the Cubical Hyperbola. 



"The plane of centres contains also three diameters, which are the 

 medians of the triangle formed by the * points ' of the section ; and 



" The locus of centres is the maximum ellipse in the triangle formed by 

 the traces of the osculating planes at the points at infinity." 



9. As a particular case of Theorem II, paragraph 4, the locus of poles 

 of planes parallel to the plane of centres is a hyperboloid of one sheet of 

 which the locus of centres is a section. The diameter conjugate to this 

 section passes through the points wi, w^, and is the locus of the " foci " of 

 planes parallel to the plane of centres. 



Again, the locus of the poles of these planes with respect to any 

 particular conic is the diameter of that conic conjugate to its diameter in 

 the plane of centres. One set of generators is therefore diameters of the 

 conic sections of the developable. Also the osculating planes touch the 

 hyperboloid. 



10. Definition. — Two osculating planes are said to correspond when the 

 centres of their sections are diametrically opposite points on the locus of 

 centres. 



Corresponding planes intersect in a line which meets the locus of centres 

 (paragraph 6), and are therefore parallel to conjugate diameters of that locus. 



They divide the line at infinity in that plane in involution, and hence 

 all " lines in two planes " are divided in involution by pairs of corresponding 

 planes. The double points are the intersections with the pair of parallel 

 osculating planes, and the centre of the involution is in the " plane of 

 centres." 



Moreover, the line of intersection of two corresponding planes intersects 

 the locus of centres, and therefore meets three generators of the same 

 system of the hyperboloid. It is accordingly also a generator. The second 

 system of generators is therefore the "lines in two corresponding planes." 



The rectangle under the distances of the points of contact of corre- 

 sponding planes from the plane of centres is constant (by the involution 

 property stated above), and therefore the ' planes ' at /3i, ^zy /Ss ai'e 

 asymptotic tangent planes to the hyperboloid. Its centre is consequently 

 the point 0. 



