78 KANSAS UNIVERSITY SCIENCE BULLETIN. 



with Us is necessarily a plane curve. We will give below a construc- 

 tion for this surface. 



11. It has been shown by Steiner( 9 ) that when a pencil of 

 planes and a pencil of quadrics are projectively related, the inter- 

 section of the corresponding surfaces generates a cubic surface. 

 It can be proved as in §5 that when the bases of the two pencils 

 have a common point, the latter is a node of the S 3. We will not 

 treat all the possible cases this way, as we doubt if the generality 

 that we have reached can thus be obtained easily. Let us consider 

 only the case where the axis of the quadrics is composed of two 

 conies, having then necessarily two common points D and D' , 

 while the axis DE of the pencil of planes goes through D, and is in 

 T fixed tangent plane of the pencil of quadrics in D. If Q 1 and Q 2 

 are the planes of the two conies, they form together one of the quad- 

 rics of the pencil, to which will correspond a plane P in the pencil 

 of planes. The lines (P, Q 1) and (P, Q 2 ) are on S 3. 



Further if 2 is the quadric corresponding to the plane P in the 

 plane-pencil, then the two lines / 1 and 1 2 of this quadric in P are 

 also on S 3. It follows that since DE is also on S 3, the latter has in 

 D a binode, since the lines on the surface through D, are in two 

 planes through DE, which is then the edge of the binode. The 

 latter is clearly a binode B 4 , since its edge DE is on S 3 . 



Suppose now that T corresponds in the plane-pencil to the system 

 (Q u Q2) in the pencil of quadrics. There T = P and the .surface 

 has in D a point U 6 . The three lines on S 3 through D are DE, 

 (T, Q 1) and (T, Q 2). Hence if in the same conditions DE coincides 

 with (T, Q 1) that is, is tangent to one of the conies in D, we have 

 the surface with a node U 7, and if finally the two conies be both 

 tangent to DE in D, then we have a surface with a node U $, the U- 

 plane of which will be the plane tangent to the quadrics osculating 

 both conies in D, or, if we prefer the plane tangent in D to a cone 

 going through both given conies. We thus have a gometric construc- 

 tion for the surface Us. 



Analytically the general equation of an S 3 with a node is 



W X 2 + X<f, ,(y.z) + 4> 3 (F *Z) = 



which can be generated by the system: 



WX + 2 -X0' 2 = O (1) 



XX = aV + bZ, if 3 = (Y + bZ)<j>' 2 , (2) 



which shows that the axis of the plane-pencil X = 0, aY-\-bZ = 

 is in X = 0, fixed tangent plane of the pencil of quadrics, and goes 

 through its contact (0, 0, 0, 1). 



9. R. Sturm. Plachen Dritter Ordnung. p. 9. 



