258 Proceedings of the Royal Irish Academy, 



Also ■ ■ 



Oi' 



ac y 





hex 



y 



a z 





c"- 



The two cables have the same " plane of centres," the same three 

 diameters, and the common points wi and wo. 



l(i. The idea of correspondence may be extended to planes other than 

 osculating planes. • 



Any plane Ax + Bi/ + Cz + D ^ contains three " points." 



hB aA D , ^ 

 The plane cc + -^— v + —z - cC = 0, 



contains the three corresponding points. These planes may be called 

 corresponding planes. 



If X' = correspond to Z = 0, it is easily verified that the 



correspondent to X + XL' = is Z - - L' = 0. 



A 



Hence " corresponding planes through a line in two corresponding planes 

 form a system in involution, the double planes being those passing through 

 the points wi, wo." 



The plane through w,, touching the cubic at wo, evidently corresponds 

 to itself, as also does the plane through wo touching the cubic at wi : hence 

 the construction for pairs of corresponding points. 



" Planes through the chord joining tuio^a, and passing through conjugate 

 diameters of the ' locus of centres ' will cut the cubic in corresponding 

 points." 



17. It is a known theorem that the anharrnonic ratio of the planes 

 through four fixed points and a variable chord is constant and equal to 

 the anharmonic ratio in which any " line in two planes " is cut by the 

 osculating planes at the fixed points. 



Taking the chord joining the points wj, wo, the anharmonic ratio is found 



to be 



si n (gi - ga) sin (g., - g^) 

 sin (g, - ga) ■ sin (g, - g,)' 



and is therefore the same for the four corresponding points. This also 

 follows from the construction given in paragraph 16. 



