CoNHAN — Some Theorems on the Twisted Cubic. 



259 



Since each of the points a>i and wa corresponds to itself, we have the 

 theorem :— 



"The points wi, (02, and any pair of corresponding points, form a harmonic 

 system." 



18. The middle point of the chord joining two corresponding points, the 



parameters of which are 0, S + ^, is 



X 3 sin (a - 2g) 2/ _ - 3 cos (a - 2g) 



or 



where 



sin 6S ' h sin 6S 



3 sin (a + Si) y 3 cos (a + SO 



sin 3 Si 'J sin 3 Si 



Si = - 2S. 



= cot 6S, 

 = - cot 3Si, 



Comparing the values in paragraph 14, we have the theorem that " The 

 locus of the middle points of chords joining corresponding points is a twisted 

 cubic which is the image in the point of the original cubic." 



Each cubic is then the locus of the middle points of chords joining 

 corresponding points on the other. 



19. The theorems in this paper are equally true with suitable modifications 

 for the cubical ellipse. There will be one diameter in the plane of centres 

 passing through the point 0. The points w„ w, will be real, and with any 

 pair of corresponding points will form a harmonic system. Corresponding 

 points will be on the same side of the " plane of centres," and their distances 

 from that plane will be connected by the relation z z ^- r. The other 

 relations are obtained by replacing l, c, and S by pure imaginaries. 



IV. 



20. Let the general equations for a twisted cubic be 



z = cj^^ + oc^- + Zed + c 



The osculating plane at t may be written 



X y z 1 



a^ h, c, d^ 



cii hi Ci dy 



ciz d> C2 dn 



^3 &3 c. d. 



f d^f' + odd- + odd + d, 



0. 



