76 KANSAS UNIVERSITY SCIENCE BULLETIN. 



therefore a common osculating plane at the limit. This construction 

 will then give the binode B 6 equivalent to %B 3 . 



If in the two preceding constructions the two cubics were sub- 

 jected to pass through another fixed point D 3 we would obtain the 

 surfaces (C 2 + B 6 ) and (6' 2 + B G ). 



The foregoing conclusions are easily verified by analysis. Let us 

 take as an example the case of B 5 . This surface has for equation ( 8 ) : 



WXZ + Y -Z + YX 2 -X 3 = (1) 



If we consider the following °° - system of quadrics: 



XY + (IX + mZ) (Y + Z) = (2) 



(X - W) (IX + mZ) - (Y - Z)Z - XW = (3) 



it represents °° 2 twisted cubics on (1), for if we eliminate (IX + mZ) 

 between (2) and (3) we obtain (1 ), while (2) and (3) have in common 

 X = Z = 0. (2) represents the projection of R 3 general cubic of the 

 system on W = 0, and as R 3 is tangent to X = Z — at the point 

 (0, 0, 0, 1), the osculating plane at the latter has for equation 

 X + (IX +mZ) = 0. From (3) we deduce that it meets R s on the 

 quadric. X °- +Z(Y — Z)— which is a cone tangent to Z = 0, in 

 which F = Z = fixed bisecant of R 3 through (0, 0, 0, 1) or D is 

 situated. Any two curves R\ and R 2 of the system considered 

 satisfy therefore the condition that they are tangent in a point D, 

 and that one of their common bisecants through D touches a quadric 

 osculating them at that same point, as we found above. 



We may remark that the plane P touching in D the quadric oscu- 

 lating R\ and R 2 is evidently one of the biplanes. 



9. If in the construction for S 3 with a binode B 4 , the cones pro- 

 jecting R\ and R 2 from the latter are tangent we obtain a binode 

 in which the biplanes coincide that is a U-node U 6 . For if D is the 

 binode, DT the tangent to both curves, the biplanes are as we have 

 seen, the planes going through DT and the two common bisecants 

 of R\ and R' 2 through D, different from OD, and they both coin- 

 cide in the case considered with a 2 . Ri and R 2 still have a bi- 

 secant a 6 going through neither D nor 0, and there is still a trans- 

 versal t as for />' 4 . If a 6 meets the C/-plane in E, DE is a line on *S 3 , 

 and finally the plane (a 6 . /) meets the plane tangent to the project- 

 ing cones of R^ and Ri along a 2 in a line g on the surface. This 

 last result is obtained by considering the limiting position of the 

 plane (a 2 , a 4 ) when a 4 approaches indefinitely a 2 . We thus have 



8. Basset Ibid. p. 61. 



