178] FREEDOM OF THE THIRD ORDER. 179 



This equation denotes a form of Steiner s surface : 



v a + V/3 + V7 + VS = 0, 

 where 



----- __ 



b c c a a o 



Q_ ^ 2y 2z 



r-* 7 7 &quot;T&quot; - j 



6 c c a a b 



2x 2y 2z 



7 = - 7 --- + ^ ------ -, + 1, 



o c c a a o 



.__ -- . 



6 c c a a 6 



From the form of its equation it appears that this surface has three 

 double lines, which meet in a point, viz. the three axes OX, OY, OZ. This 

 being so any plane will cut the surface in a quartic curve with three double 

 points, being those in which the plane cuts the axes. If the plane touch the 

 surface, the point of contact is an additional double point on the section, that 

 is, the section will be a quartic curve with four double points, i.e. a pair of 

 conies. The projections of the origin on the generators of any cylindroid 

 belonging to the system lie on a plane ellipse ( 23). This ellipse must lie 

 on the Steiner quartic. Hence the plane of the ellipse must cut the quartic 

 in two conies and must be a tangent plane. See note on p. 182. 



178. Screws of the Three-System parallel to a Plane. 



Up to the present we have been analysing the screw system by classifying 

 the screws into groups of constant pitch. Some interesting features will be 

 presented by adopting a new method of classification. We shall now divide 

 the general system into groups of screws which are parallel to the same 

 plane. 



We shall first prove that each of these groups is in general a cylindroid. 

 For suppose a screw of infinite pitch normal to the plane, then all the screws 

 of the group parallel to the plane are reciprocal to this screw of infinite 

 pitch. But they are also reciprocal to any three screws of the original 

 reciprocal system ; they, therefore, form a screw system of the second order 

 ( 72) that is, they constitute a cylindroid. 



We shall prove this in another manner. 



A quadric containing a line must touch every plane passing through the 

 line. The number of screws of the system which can lie in a given plane 

 is, therefore, equal to the number of the quadrics of the system which can 

 be drawn to touch that plane. 



122 



