CoNRAN — Some Theorems on the Tiuisted Cuhic. 



253 



7. The case in which the bitangent is at infinity is of particular 

 importance. 



The cuspidal tangents become the medians of the triangle formed by the 

 cusps ; and the conic becomes the maximum ellipse in a similar and similarly 

 placed triangle of f the linear dimensions of the first. 



II. 



8. The section of the developable by an osculating plane being the " line " 

 counted twice and a conic touching the " line " at the " point," it is clear from 

 fig. 1 that if any plane pass through the line a, then y is its pole with 

 respect to one conic, and the line V contains its pole with respect to the 

 second conic. 



Now, let ai, 02, «3 be the " points " in any plane P, and ai^i, a^oh, a^^h the 

 cuspidal tangents to the section of the developable meeting the section again 

 in «!, Oh, Oh. 



Let the osculating planes at ai, a-i, a^ be Ai, A2, and A3. 



Let /3i, (^2, jSj be the points of contact of tangents to the cubic from 

 Xi,cc2,o:3, and yi, ^2, ?/3 the points where the " planes" -5j, ^2,-^3 are met by the 

 " lines " at oi, 02, 03. The poles of the plane P with respect to the sections 

 Ai, A2, A3 are 3/1, 3/2, 2/3, and the poles with respect to the sections £1, B2, Pz 

 are in the lines /Si^/i, j32z/2, and jSs^/s respectively. 



P)3 fX3 



Fig. 3. Fig. 4. 



Hence, the lines j3i2/i, /Ba^/a, ^zVi are in the plane of the locus of poles 

 of the plane P. Denoting this plane by Q, it is clear from Theorem III, 

 paragraph 4, that P and Q are " conjoint planes " (Cremona, CreUe, vol. 58), 

 and that they intersect in a " line in two planes." 



